Theorem cgsex2g 3239

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 501	. . 3 |- ( ( ch /\ ph ) -> ps )
3	2	exlimivv 1860	. 2 |- ( E. x E. y ( ch /\ ph ) -> ps )
4		elisset 3215	. . . . . 6 |- ( A e. V -> E. x x = A )
5		elisset 3215	. . . . . 6 |- ( B e. W -> E. y y = B )
6	4, 5	anim12i 590	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( E. x x = A /\ E. y y = B ) )
7		eeanv 2182	. . . . 5 |- ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
8	6, 7	sylibr 224	. . . 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 1763	. . . 4 |- ( E. x E. y ( x = A /\ y = B ) -> E. x E. y ch )
11	8, 10	syl 17	. . 3 |- ( ( A e. V /\ B e. W ) -> E. x E. y ch )
12	1	biimprcd 240	. . . . 5 |- ( ps -> ( ch -> ph ) )
13	12	ancld 576	. . . 4 |- ( ps -> ( ch -> ( ch /\ ph ) ) )
14	13	2eximdv 1848	. . 3 |- ( ps -> ( E. x E. y ch -> E. x E. y ( ch /\ ph ) ) )
15	11, 14	syl5com 31	. 2 |- ( ( A e. V /\ B e. W ) -> ( ps -> E. x E. y ( ch /\ ph ) ) )
16	3, 15	impbid2 216	1 |- ( ( A e. V /\ B e. W ) -> ( E. x E. y ( 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-10 2019  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by: (None)