Theorem cgsex2g 2766

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 294	. . 3 |- ( ( ch /\ ph ) -> ps )
3	2	exlimivv 1889	. 2 |- ( E. x E. y ( ch /\ ph ) -> ps )
4		elisset 2744	. . . . . 6 |- ( A e. V -> E. x x = A )
5		elisset 2744	. . . . . 6 |- ( B e. W -> E. y y = B )
6	4, 5	anim12i 336	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( E. x x = A /\ E. y y = B ) )
7		eeanv 1925	. . . . 5 |- ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
8	6, 7	sylibr 133	. . . 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 1594	. . . 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 159	. . . . 5 |- ( ps -> ( ch -> ph ) )
13	12	ancld 323	. . . 4 |- ( ps -> ( ch -> ( ch /\ ph ) ) )
14	13	2eximdv 1875	. . 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 142	1 |- ( ( A e. V /\ B e. W ) -> ( E. x E. y ( ch /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   E.wex 1485   e. wcel 2141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by: (None)