Theorem cdleme29b

Description: Transform cdleme28 . (Compare cdleme25b .) TODO: FIX COMMENT. (Contributed by NM, 7-Feb-2013)

	Ref	Expression
Hypotheses	cdleme26.b	$$⊢B={\mathrm{Base}}_K$$
	cdleme26.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme26.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme26.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme26.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme26.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme27.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme27.f	$$⊢F=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme27.z	$$⊢Z=\left(z\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme27.n	$$⊢N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme27.d	$$⊢D=\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\right)$$
	cdleme27.c	$$⊢C=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),D,F\right)$$
Assertion	cdleme29b	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge P\ne Q\wedge \left(X\in B\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\to \exists v\in B\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to v=C\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdleme26.b	$$⊢B={\mathrm{Base}}_K$$
2		cdleme26.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdleme26.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdleme26.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdleme26.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdleme26.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdleme27.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		cdleme27.f	$$⊢F=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme27.z	$$⊢Z=\left(z\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
10		cdleme27.n	$$⊢N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
11		cdleme27.d	$$⊢D=\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\right)$$
12		cdleme27.c	$$⊢C=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),D,F\right)$$
13	1 2 3 4 5 6 7 8 9 10 11 12	cdleme29ex	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge P\ne Q\wedge \left(X\in B\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\to \exists s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\wedge C\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\in B\right)$$
14		eqid	$$⊢\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
15		eqid	$$⊢\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
16		eqid	$$⊢\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)=\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)$$
17		eqid	$$⊢if\left(t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right),\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)=if\left(t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right),\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)$$
18	1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 17	cdleme28	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge P\ne Q\wedge \left(X\in B\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\to \forall s\in A\forall t\in A\left(\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\wedge \left(\neg t\underset{\dot{}}{\le }W\wedge t\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\to C\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=if\left(t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right),\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)$$
19		breq1	$$⊢s=t\to \left(s\underset{\dot{}}{\le }W\leftrightarrow t\underset{\dot{}}{\le }W\right)$$
20	19	notbid	$$⊢s=t\to \left(\neg s\underset{\dot{}}{\le }W\leftrightarrow \neg t\underset{\dot{}}{\le }W\right)$$
21		oveq1	$$⊢s=t\to s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=t\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)$$
22	21	eqeq1d	$$⊢s=t\to \left(s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\leftrightarrow t\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)$$
23	20 22	anbi12d	$$⊢s=t\to \left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\leftrightarrow \left(\neg t\underset{\dot{}}{\le }W\wedge t\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)$$
24	12	oveq1i	$$⊢C\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),D,F\right)\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)$$
25		breq1	$$⊢s=t\to \left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\leftrightarrow t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
26		oveq1	$$⊢s=t\to s\underset{\dot{}}{\vee }z=t\underset{\dot{}}{\vee }z$$
27	26	oveq1d	$$⊢s=t\to \left(s\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W=\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W$$
28	27	oveq2d	$$⊢s=t\to Z\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)=Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)$$
29	28	oveq2d	$$⊢s=t\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
30	10 29	eqtrid	$$⊢s=t\to N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
31	30	eqeq2d	$$⊢s=t\to \left(u=N\leftrightarrow u=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)\right)$$
32	31	imbi2d	$$⊢s=t\to \left(\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\leftrightarrow \left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)$$
33	32	ralbidv	$$⊢s=t\to \left(\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\leftrightarrow \forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)$$
34	33	riotabidv	$$⊢s=t\to \left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\right)=\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)$$
35	11 34	eqtrid	$$⊢s=t\to D=\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)$$
36		oveq1	$$⊢s=t\to s\underset{\dot{}}{\vee }U=t\underset{\dot{}}{\vee }U$$
37		oveq2	$$⊢s=t\to P\underset{\dot{}}{\vee }s=P\underset{\dot{}}{\vee }t$$
38	37	oveq1d	$$⊢s=t\to \left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W=\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W$$
39	38	oveq2d	$$⊢s=t\to Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)=Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)$$
40	36 39	oveq12d	$$⊢s=t\to \left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
41	8 40	eqtrid	$$⊢s=t\to F=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
42	25 35 41	ifbieq12d	$$⊢s=t\to if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),D,F\right)=if\left(t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right),\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)$$
43	42	oveq1d	$$⊢s=t\to if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),D,F\right)\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=if\left(t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right),\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)$$
44	24 43	eqtrid	$$⊢s=t\to C\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=if\left(t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right),\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)$$
45	23 44	reusv3	$$⊢\exists s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\wedge C\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\in B\right)\to \left(\forall s\in A\forall t\in A\left(\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\wedge \left(\neg t\underset{\dot{}}{\le }W\wedge t\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\to C\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=if\left(t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right),\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\leftrightarrow \exists v\in B\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to v=C\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\right)$$
46	45	biimpd	$$⊢\exists s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\wedge C\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\in B\right)\to \left(\forall s\in A\forall t\in A\left(\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\wedge \left(\neg t\underset{\dot{}}{\le }W\wedge t\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\to C\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=if\left(t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right),\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\to \exists v\in B\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to v=C\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\right)$$
47	13 18 46	sylc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge P\ne Q\wedge \left(X\in B\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\to \exists v\in B\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to v=C\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)$$