Theorem cgsexg 2589

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 280	. . 3 |- ( ( ch /\ ph ) -> ps )
3	2	exlimiv 1489	. 2 |- ( E. x ( ch /\ ph ) -> ps )
4		elisset 2568	. . . 4 |- ( A e. V -> E. x x = A )
5		cgsexg.1	. . . . 5 |- ( x = A -> ch )
6	5	eximi 1491	. . . 4 |- ( E. x x = A -> E. x ch )
7	4, 6	syl 14	. . 3 |- ( A e. V -> E. x ch )
8	1	biimprcd 149	. . . . 5 |- ( ps -> ( ch -> ph ) )
9	8	ancld 308	. . . 4 |- ( ps -> ( ch -> ( ch /\ ph ) ) )
10	9	eximdv 1760	. . 3 |- ( ps -> ( E. x ch -> E. x ( ch /\ ph ) ) )
11	7, 10	syl5com 26	. 2 |- ( A e. V -> ( ps -> E. x ( ch /\ ph ) ) )
12	3, 11	impbid2 131	1 |- ( A e. V -> ( E. x ( ch /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by: (None)