Theorem cdleme40v

Description: Part of proof of Lemma E in Crawley p. 113. Change bound variables in [_ S / u ]_ V (but we use [_ R / u ]_ V for convenience since we have its hypotheses available). (Contributed by NM, 18-Mar-2013)

	Ref	Expression
Hypotheses	cdleme40.b	$$⊢B={\mathrm{Base}}_K$$
	cdleme40.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme40.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme40.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme40.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme40.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme40.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme40.e	$$⊢E=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme40.g	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme40.i	$$⊢I=\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=G\right)\right)$$
	cdleme40.n	$$⊢N=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),I,D\right)$$
	cdleme40.d	$$⊢D=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme40r.y	$$⊢Y=\left(u\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }u\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme40r.t	$$⊢T=\left(v\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme40r.x	$$⊢X=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme40r.o	$$⊢O=\left(\iota z\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to z=X\right)\right)$$
	cdleme40r.v	$$⊢V=if\left(u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),O,Y\right)$$
Assertion	cdleme40v	$$⊢R\in A\to ⦋R/s⦌N=⦋R/u⦌V$$

Proof

Step	Hyp	Ref	Expression
1		cdleme40.b	$$⊢B={\mathrm{Base}}_K$$
2		cdleme40.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdleme40.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdleme40.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdleme40.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdleme40.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdleme40.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		cdleme40.e	$$⊢E=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme40.g	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
10		cdleme40.i	$$⊢I=\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=G\right)\right)$$
11		cdleme40.n	$$⊢N=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),I,D\right)$$
12		cdleme40.d	$$⊢D=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
13		cdleme40r.y	$$⊢Y=\left(u\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }u\right)\underset{\dot{}}{\wedge }W\right)\right)$$
14		cdleme40r.t	$$⊢T=\left(v\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
15		cdleme40r.x	$$⊢X=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
16		cdleme40r.o	$$⊢O=\left(\iota z\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to z=X\right)\right)$$
17		cdleme40r.v	$$⊢V=if\left(u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),O,Y\right)$$
18		breq1	$$⊢s=u\to \left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\leftrightarrow u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
19		oveq1	$$⊢s=u\to s\underset{\dot{}}{\vee }t=u\underset{\dot{}}{\vee }t$$
20	19	oveq1d	$$⊢s=u\to \left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W=\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W$$
21	20	oveq2d	$$⊢s=u\to E\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)=E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)$$
22	21	oveq2d	$$⊢s=u\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
23	9 22	syl5eq	$$⊢s=u\to G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
24	23	eqeq2d	$$⊢s=u\to \left(y=G\leftrightarrow y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)$$
25	24	imbi2d	$$⊢s=u\to \left(\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=G\right)\leftrightarrow \left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)$$
26	25	ralbidv	$$⊢s=u\to \left(\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=G\right)\leftrightarrow \forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)$$
27	26	riotabidv	$$⊢s=u\to \left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=G\right)\right)=\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)$$
28		eqeq1	$$⊢y=z\to \left(y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\leftrightarrow z=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)$$
29	28	imbi2d	$$⊢y=z\to \left(\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\leftrightarrow \left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to z=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)$$
30	29	ralbidv	$$⊢y=z\to \left(\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\leftrightarrow \forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to z=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)$$
31		breq1	$$⊢t=v\to \left(t\underset{\dot{}}{\le }W\leftrightarrow v\underset{\dot{}}{\le }W\right)$$
32	31	notbid	$$⊢t=v\to \left(\neg t\underset{\dot{}}{\le }W\leftrightarrow \neg v\underset{\dot{}}{\le }W\right)$$
33		breq1	$$⊢t=v\to \left(t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\leftrightarrow v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
34	33	notbid	$$⊢t=v\to \left(\neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\leftrightarrow \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
35	32 34	anbi12d	$$⊢t=v\to \left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\leftrightarrow \left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)$$
36		oveq1	$$⊢t=v\to t\underset{\dot{}}{\vee }U=v\underset{\dot{}}{\vee }U$$
37		oveq2	$$⊢t=v\to P\underset{\dot{}}{\vee }t=P\underset{\dot{}}{\vee }v$$
38	37	oveq1d	$$⊢t=v\to \left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W=\left(P\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W$$
39	38	oveq2d	$$⊢t=v\to Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)=Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)$$
40	36 39	oveq12d	$$⊢t=v\to \left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(v\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
41	40 8 14	3eqtr4g	$$⊢t=v\to E=T$$
42		oveq2	$$⊢t=v\to u\underset{\dot{}}{\vee }t=u\underset{\dot{}}{\vee }v$$
43	42	oveq1d	$$⊢t=v\to \left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W=\left(u\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W$$
44	41 43	oveq12d	$$⊢t=v\to E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)=T\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)$$
45	44	oveq2d	$$⊢t=v\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
46	45 15	eqtr4di	$$⊢t=v\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)=X$$
47	46	eqeq2d	$$⊢t=v\to \left(z=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\leftrightarrow z=X\right)$$
48	35 47	imbi12d	$$⊢t=v\to \left(\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to z=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\leftrightarrow \left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to z=X\right)\right)$$
49	48	cbvralvw	$$⊢\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to z=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\leftrightarrow \forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to z=X\right)$$
50	30 49	bitrdi	$$⊢y=z\to \left(\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\leftrightarrow \forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to z=X\right)\right)$$
51	50	cbvriotavw	$$⊢\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)=\left(\iota z\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to z=X\right)\right)$$
52	27 51	eqtrdi	$$⊢s=u\to \left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=G\right)\right)=\left(\iota z\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to z=X\right)\right)$$
53	52 10 16	3eqtr4g	$$⊢s=u\to I=O$$
54		oveq1	$$⊢s=u\to s\underset{\dot{}}{\vee }U=u\underset{\dot{}}{\vee }U$$
55		oveq2	$$⊢s=u\to P\underset{\dot{}}{\vee }s=P\underset{\dot{}}{\vee }u$$
56	55	oveq1d	$$⊢s=u\to \left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W=\left(P\underset{\dot{}}{\vee }u\right)\underset{\dot{}}{\wedge }W$$
57	56	oveq2d	$$⊢s=u\to Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)=Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }u\right)\underset{\dot{}}{\wedge }W\right)$$
58	54 57	oveq12d	$$⊢s=u\to \left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(u\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }u\right)\underset{\dot{}}{\wedge }W\right)\right)$$
59	58 12 13	3eqtr4g	$$⊢s=u\to D=Y$$
60	18 53 59	ifbieq12d	$$⊢s=u\to if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),I,D\right)=if\left(u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),O,Y\right)$$
61	60 11 17	3eqtr4g	$$⊢s=u\to N=V$$
62	61	cbvcsbv	$$⊢⦋R/s⦌N=⦋R/u⦌V$$
63	62	a1i	$$⊢R\in A\to ⦋R/s⦌N=⦋R/u⦌V$$