Theorem spc2gv 2697

Description: Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)

Hypothesis

Ref	Expression
spc2egv.1	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
spc2gv	|- ( ( A e. V /\ B e. W ) -> ( A. x A. y ph -> ps ) )

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

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

Proof of Theorem spc2gv

Step	Hyp	Ref	Expression
1		elisset 2622	. . . 4 |- ( A e. V -> E. x x = A )
2		elisset 2622	. . . 4 |- ( B e. W -> E. y y = B )
3	1, 2	anim12i 331	. . 3 |- ( ( A e. V /\ B e. W ) -> ( E. x x = A /\ E. y y = B ) )
4		eeanv 1850	. . 3 |- ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
5	3, 4	sylibr 132	. 2 |- ( ( A e. V /\ B e. W ) -> E. x E. y ( x = A /\ y = B ) )
6		spc2egv.1	. . . . . 6 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
7	6	biimpcd 157	. . . . 5 |- ( ph -> ( ( x = A /\ y = B ) -> ps ) )
8	7	2alimi 1386	. . . 4 |- ( A. x A. y ph -> A. x A. y ( ( x = A /\ y = B ) -> ps ) )
9		exim 1531	. . . . 5 |- ( A. y ( ( x = A /\ y = B ) -> ps ) -> ( E. y ( x = A /\ y = B ) -> E. y ps ) )
10	9	alimi 1385	. . . 4 |- ( A. x A. y ( ( x = A /\ y = B ) -> ps ) -> A. x ( E. y ( x = A /\ y = B ) -> E. y ps ) )
11		exim 1531	. . . 4 |- ( A. x ( E. y ( x = A /\ y = B ) -> E. y ps ) -> ( E. x E. y ( x = A /\ y = B ) -> E. x E. y ps ) )
12	8, 10, 11	3syl 17	. . 3 |- ( A. x A. y ph -> ( E. x E. y ( x = A /\ y = B ) -> E. x E. y ps ) )
13		19.9v 1794	. . . 4 |- ( E. x E. y ps <-> E. y ps )
14		19.9v 1794	. . . 4 |- ( E. y ps <-> ps )
15	13, 14	bitri 182	. . 3 |- ( E. x E. y ps <-> ps )
16	12, 15	syl6ib 159	. 2 |- ( A. x A. y ph -> ( E. x E. y ( x = A /\ y = B ) -> ps ) )
17	5, 16	syl5com 29	1 |- ( ( A e. V /\ B e. W ) -> ( A. x A. y ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1283   = wceq 1285   E.wex 1422   e. wcel 1434

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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-v 2612

This theorem is referenced by:  rspc2gv  2720  trel  3902  exmidundif  3991  elovmpt2  5752