Theorem cgsexg

Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007)

	Ref	Expression
Hypotheses	cgsexg.1	|- ( x = A -> ch )
	cgsexg.2	|- ( ch -> ( ph <-> ps ) )
Assertion	cgsexg	|- ( A e. V -> ( E. x ( ch /\ ph ) <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		cgsexg.1	|- ( x = A -> ch )
2		cgsexg.2	|- ( ch -> ( ph <-> ps ) )
3	2	biimpa	|- ( ( ch /\ ph ) -> ps )
4	3	exlimiv	|- ( E. x ( ch /\ ph ) -> ps )
5		elisset	|- ( A e. V -> E. x x = A )
6	1	eximi	|- ( E. x x = A -> E. x ch )
7	5 6	syl	|- ( A e. V -> E. x ch )
8	2	biimprcd	|- ( ps -> ( ch -> ph ) )
9	8	ancld	|- ( ps -> ( ch -> ( ch /\ ph ) ) )
10	9	eximdv	|- ( ps -> ( E. x ch -> E. x ( ch /\ ph ) ) )
11	7 10	syl5com	|- ( A e. V -> ( ps -> E. x ( ch /\ ph ) ) )
12	4 11	impbid2	|- ( A e. V -> ( E. x ( ch /\ ph ) <-> ps ) )