Theorem cdleme31fv1s

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 25-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	$$⊢F=\left(x\in B⟼if\left(\left(P\ne Q\wedge \neg x\underset{\dot{}}{\le }W\right),O,x\right)\right)$$
Assertion	cdleme31fv1s	$$⊢\left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\to F\left(X\right)=⦋X/x⦌O$$

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		eqid	$$⊢\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to z=N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\right)=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to z=N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\right)$$
4	1 2 3	cdleme31fv1	$$⊢\left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\to F\left(X\right)=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to z=N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\right)$$
5	1 3	cdleme31so	$$⊢X\in B\to ⦋X/x⦌O=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to z=N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\right)$$
6	5	adantr	$$⊢\left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\to ⦋X/x⦌O=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to z=N\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\right)$$
7	4 6	eqtr4d	$$⊢\left(X\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\to F\left(X\right)=⦋X/x⦌O$$