Theorem cdleme31fv

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 10-Feb-2013)

	Ref	Expression
Hypotheses	cdleme31.o	$$⊢O=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=N\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right)$$
	cdleme31.f	$$⊢F=\left(x\in B⟼if\left(\left(P\ne Q\wedge \neg x\underset{\dot{}}{\le }W\right),O,x\right)\right)$$
	cdleme31.c	$$⊢C=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to z=N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\right)$$
Assertion	cdleme31fv	$$⊢X\in B\to F\left(X\right)=if\left(\left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right),C,X\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdleme31.o	$$⊢O=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=N\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right)$$
2		cdleme31.f	$$⊢F=\left(x\in B⟼if\left(\left(P\ne Q\wedge \neg x\underset{\dot{}}{\le }W\right),O,x\right)\right)$$
3		cdleme31.c	$$⊢C=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to z=N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\right)$$
4		riotaex	$$⊢\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to z=N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\right)\in \mathrm{V}$$
5	3 4	eqeltri	$$⊢C\in \mathrm{V}$$
6		ifexg	$$⊢\left(C\in \mathrm{V}\wedge X\in B\right)\to if\left(\left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right),C,X\right)\in \mathrm{V}$$
7	5 6	mpan	$$⊢X\in B\to if\left(\left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right),C,X\right)\in \mathrm{V}$$
8		breq1	$$⊢x=X\to \left(x\underset{\dot{}}{\le }W\leftrightarrow X\underset{\dot{}}{\le }W\right)$$
9	8	notbid	$$⊢x=X\to \left(\neg x\underset{\dot{}}{\le }W\leftrightarrow \neg X\underset{\dot{}}{\le }W\right)$$
10	9	anbi2d	$$⊢x=X\to \left(\left(P\ne Q\wedge \neg x\underset{\dot{}}{\le }W\right)\leftrightarrow \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)$$
11		oveq1	$$⊢x=X\to x\underset{\dot{}}{\wedge }W=X\underset{\dot{}}{\wedge }W$$
12	11	oveq2d	$$⊢x=X\to s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)$$
13		id	$$⊢x=X\to x=X$$
14	12 13	eqeq12d	$$⊢x=X\to \left(s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\leftrightarrow s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)$$
15	14	anbi2d	$$⊢x=X\to \left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\leftrightarrow \left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)$$
16	11	oveq2d	$$⊢x=X\to N\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)$$
17	16	eqeq2d	$$⊢x=X\to \left(z=N\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\leftrightarrow z=N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)$$
18	15 17	imbi12d	$$⊢x=X\to \left(\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=N\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\leftrightarrow \left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to z=N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\right)$$
19	18	ralbidv	$$⊢x=X\to \left(\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=N\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\leftrightarrow \forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to z=N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\right)$$
20	19	riotabidv	$$⊢x=X\to \left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=N\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right)=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to z=N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\right)$$
21	20 1 3	3eqtr4g	$$⊢x=X\to O=C$$
22	10 21 13	ifbieq12d	$$⊢x=X\to if\left(\left(P\ne Q\wedge \neg x\underset{\dot{}}{\le }W\right),O,x\right)=if\left(\left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right),C,X\right)$$
23	22 2	fvmptg	$$⊢\left(X\in B\wedge if\left(\left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right),C,X\right)\in \mathrm{V}\right)\to F\left(X\right)=if\left(\left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right),C,X\right)$$
24	7 23	mpdan	$$⊢X\in B\to F\left(X\right)=if\left(\left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right),C,X\right)$$