Theorem spc2egv 2802

Description: Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)

Hypothesis

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

Assertion

Ref	Expression
spc2egv	|- ( ( A e. V /\ B e. W ) -> ( ps -> E. x E. y ph ) )

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 spc2egv

Step	Hyp	Ref	Expression
1		elisset 2726	. . . 4 |- ( A e. V -> E. x x = A )
2		elisset 2726	. . . 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 1912	. . 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	. . . 4 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
7	6	biimprcd 159	. . 3 |- ( ps -> ( ( x = A /\ y = B ) -> ph ) )
8	7	2eximdv 1862	. 2 |- ( ps -> ( E. x E. y ( x = A /\ y = B ) -> E. x E. y ph ) )
9	5, 8	syl5com 29	1 |- ( ( A e. V /\ B e. W ) -> ( ps -> E. x E. y ph ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   E.wex 1472   e. wcel 2128

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714

This theorem is referenced by:  spc2ev  2808  th3q  6587  addnnnq0  7371  mulnnnq0  7372  addsrpr  7667  mulsrpr  7668