Theorem cgsexg 3238

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 501	. . 3 |- ( ( ch /\ ph ) -> ps )
3	2	exlimiv 1858	. 2 |- ( E. x ( ch /\ ph ) -> ps )
4		elisset 3215	. . . 4 |- ( A e. V -> E. x x = A )
5		cgsexg.1	. . . . 5 |- ( x = A -> ch )
6	5	eximi 1762	. . . 4 |- ( E. x x = A -> E. x ch )
7	4, 6	syl 17	. . 3 |- ( A e. V -> E. x ch )
8	1	biimprcd 240	. . . . 5 |- ( ps -> ( ch -> ph ) )
9	8	ancld 576	. . . 4 |- ( ps -> ( ch -> ( ch /\ ph ) ) )
10	9	eximdv 1846	. . 3 |- ( ps -> ( E. x ch -> E. x ( ch /\ ph ) ) )
11	7, 10	syl5com 31	. 2 |- ( A e. V -> ( ps -> E. x ( ch /\ ph ) ) )
12	3, 11	impbid2 216	1 |- ( A e. V -> ( E. x ( ch /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by: (None)