Theorem cgsexg 2798

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

Hypotheses

Ref	Expression
cgsexg.1	|- ( x = A -> ch )
cgsexg.2	|- ( ch -> ( ph <-> ps ) )

Assertion

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

Distinct variable groups:   x, A   ps, x

Allowed substitution hints:   ph( x)   ch( x)   V( x)

Proof of Theorem cgsexg

Step	Hyp	Ref	Expression
1		cgsexg.2	. . . 4 |- ( ch -> ( ph <-> ps ) )
2	1	biimpa 296	. . 3 |- ( ( ch /\ ph ) -> ps )
3	2	exlimiv 1612	. 2 |- ( E. x ( ch /\ ph ) -> ps )
4		elisset 2777	. . . 4 |- ( A e. V -> E. x x = A )
5		cgsexg.1	. . . . 5 |- ( x = A -> ch )
6	5	eximi 1614	. . . 4 |- ( E. x x = A -> E. x ch )
7	4, 6	syl 14	. . 3 |- ( A e. V -> E. x ch )
8	1	biimprcd 160	. . . . 5 |- ( ps -> ( ch -> ph ) )
9	8	ancld 325	. . . 4 |- ( ps -> ( ch -> ( ch /\ ph ) ) )
10	9	eximdv 1894	. . 3 |- ( ps -> ( E. x ch -> E. x ( ch /\ ph ) ) )
11	7, 10	syl5com 29	. 2 |- ( A e. V -> ( ps -> E. x ( ch /\ ph ) ) )
12	3, 11	impbid2 143	1 |- ( A e. V -> ( E. x ( ch /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1506   e. wcel 2167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-v 2765

This theorem is referenced by: (None)