Theorem cdleme15c

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph on p. 114, showing, in their notation, ((p \/ s_1) /\ (q \/ s_1)) \/ ((p \/ t_1) /\ (q \/ t_1))=s_1 \/ t_1. C and X represent s_1 and t_1 respectively. The order of our operations is slightly different. (Contributed by NM, 10-Oct-2012)

	Ref	Expression
Hypotheses	cdleme12.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme12.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme12.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme12.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme12.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme12.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme12.f	$$⊢F=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme12.g	$$⊢G=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme15.c	$$⊢C=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
	cdleme15.x	$$⊢X=\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
Assertion	cdleme15c	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }X\right)\right)\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }C\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }C\right)\right)=X\underset{\dot{}}{\vee }C$$

Proof

Step	Hyp	Ref	Expression
1		cdleme12.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme12.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme12.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme12.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme12.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme12.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7		cdleme12.f	$$⊢F=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
8		cdleme12.g	$$⊢G=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme15.c	$$⊢C=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
10		cdleme15.x	$$⊢X=\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
11		simp11	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
12		simp12	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
13		simp13	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)$$
14		simp22	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)$$
15		simp21	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)$$
16		simp23l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to P\ne Q$$
17		simp23r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to S\ne T$$
18	17	necomd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to T\ne S$$
19	16 18	jca	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(P\ne Q\wedge T\ne S\right)$$
20		simp32	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
21		simp31	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
22		simp33	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)$$
23		simp11l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to K\in \mathrm{HL}$$
24		simp21l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to S\in A$$
25		simp22l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to T\in A$$
26	2 4	hlatjcom	$$⊢\left(K\in \mathrm{HL}\wedge S\in A\wedge T\in A\right)\to S\underset{\dot{}}{\vee }T=T\underset{\dot{}}{\vee }S$$
27	23 24 25 26	syl3anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to S\underset{\dot{}}{\vee }T=T\underset{\dot{}}{\vee }S$$
28	27	breq2d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\leftrightarrow U\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }S\right)\right)$$
29	22 28	mtbid	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \neg U\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }S\right)$$
30	1 2 3 4 5 6 8 7 10 9	cdleme15b	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge T\ne S\right)\right)\wedge \left(\neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }S\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }X\right)=X$$
31	11 12 13 14 15 19 20 21 29 30	syl333anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }X\right)=X$$
32	1 2 3 4 5 6 7 8 9 10	cdleme15b	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }C\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }C\right)=C$$
33	31 32	oveq12d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg T\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge S\ne T\right)\right)\wedge \left(\neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg U\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }X\right)\right)\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }C\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }C\right)\right)=X\underset{\dot{}}{\vee }C$$