Theorem spc2gv 2780

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 2703	. . . 4 |- ( A e. V -> E. x x = A )
2		elisset 2703	. . . 4 |- ( B e. W -> E. y y = B )
3	1, 2	anim12i 336	. . 3 |- ( ( A e. V /\ B e. W ) -> ( E. x x = A /\ E. y y = B ) )
4		eeanv 1905	. . 3 |- ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
5	3, 4	sylibr 133	. 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 158	. . . . 5 |- ( ph -> ( ( x = A /\ y = B ) -> ps ) )
8	7	2alimi 1433	. . . 4 |- ( A. x A. y ph -> A. x A. y ( ( x = A /\ y = B ) -> ps ) )
9		exim 1579	. . . . 5 |- ( A. y ( ( x = A /\ y = B ) -> ps ) -> ( E. y ( x = A /\ y = B ) -> E. y ps ) )
10	9	alimi 1432	. . . 4 |- ( A. x A. y ( ( x = A /\ y = B ) -> ps ) -> A. x ( E. y ( x = A /\ y = B ) -> E. y ps ) )
11		exim 1579	. . . 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 1844	. . . 4 |- ( E. x E. y ps <-> E. y ps )
14		19.9v 1844	. . . 4 |- ( E. y ps <-> ps )
15	13, 14	bitri 183	. . 3 |- ( E. x E. y ps <-> ps )
16	12, 15	syl6ib 160	. 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 103   <-> wb 104   A.wal 1330   = wceq 1332   E.wex 1469   e. wcel 1481

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691

This theorem is referenced by:  rspc2gv  2805  trel  4041  exmidundif  4137  exmidundifim  4138  elovmpo  5979  cnmpt12  12495  cnmpt22  12502  exmidsbthrlem  13392