Theorem cgsex2g 2590

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 280	. . 3 |- ( ( ch /\ ph ) -> ps )
3	2	exlimivv 1776	. 2 |- ( E. x E. y ( ch /\ ph ) -> ps )
4		elisset 2568	. . . . . 6 |- ( A e. V -> E. x x = A )
5		elisset 2568	. . . . . 6 |- ( B e. W -> E. y y = B )
6	4, 5	anim12i 321	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( E. x x = A /\ E. y y = B ) )
7		eeanv 1807	. . . . 5 |- ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
8	6, 7	sylibr 137	. . . 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 1492	. . . 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 149	. . . . 5 |- ( ps -> ( ch -> ph ) )
13	12	ancld 308	. . . 4 |- ( ps -> ( ch -> ( ch /\ ph ) ) )
14	13	2eximdv 1762	. . 3 |- ( ps -> ( E. x E. y ch -> E. x E. y ( ch /\ ph ) ) )
15	11, 14	syl5com 26	. 2 |- ( ( A e. V /\ B e. W ) -> ( ps -> E. x E. y ( ch /\ ph ) ) )
16	3, 15	impbid2 131	1 |- ( ( A e. V /\ B e. W ) -> ( E. x E. y ( 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-7 1337  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-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by: (None)