Theorem cgsex2g 2635

Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)

Hypotheses

Ref	Expression
cgsex2g.1	|- ( ( x = A /\ y = B ) -> ch )
cgsex2g.2	|- ( ch -> ( ph <-> ps ) )

Assertion

Ref	Expression
cgsex2g	|- ( ( A e. V /\ B e. W ) -> ( E. x E. y ( ch /\ ph ) <-> ps ) )

Distinct variable groups:   x, y, ps   x, A, y   x, B, y

Allowed substitution hints:   ph( x, y)   ch( x, y)   V( x, y)   W( x, y)

Proof of Theorem cgsex2g

Step	Hyp	Ref	Expression
1		cgsex2g.2	. . . 4 |- ( ch -> ( ph <-> ps ) )
2	1	biimpa 290	. . 3 |- ( ( ch /\ ph ) -> ps )
3	2	exlimivv 1817	. 2 |- ( E. x E. y ( ch /\ ph ) -> ps )
4		elisset 2613	. . . . . 6 |- ( A e. V -> E. x x = A )
5		elisset 2613	. . . . . 6 |- ( B e. W -> E. y y = B )
6	4, 5	anim12i 331	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( E. x x = A /\ E. y y = B ) )
7		eeanv 1848	. . . . 5 |- ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
8	6, 7	sylibr 132	. . . 4 |- ( ( A e. V /\ B e. W ) -> E. x E. y ( x = A /\ y = B ) )
9		cgsex2g.1	. . . . 5 |- ( ( x = A /\ y = B ) -> ch )
10	9	2eximi 1532	. . . 4 |- ( E. x E. y ( x = A /\ y = B ) -> E. x E. y ch )
11	8, 10	syl 14	. . 3 |- ( ( A e. V /\ B e. W ) -> E. x E. y ch )
12	1	biimprcd 158	. . . . 5 |- ( ps -> ( ch -> ph ) )
13	12	ancld 318	. . . 4 |- ( ps -> ( ch -> ( ch /\ ph ) ) )
14	13	2eximdv 1803	. . 3 |- ( ps -> ( E. x E. y ch -> E. x E. y ( ch /\ ph ) ) )
15	11, 14	syl5com 29	. 2 |- ( ( A e. V /\ B e. W ) -> ( ps -> E. x E. y ( ch /\ ph ) ) )
16	3, 15	impbid2 141	1 |- ( ( A e. V /\ B e. W ) -> ( E. x E. y ( ch /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by: (None)