Theorem spc2egv 2820

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 2744	. . . 4 |- ( A e. V -> E. x x = A )
2		elisset 2744	. . . 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 1925	. . 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 1875	. 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 1348   E.wex 1485   e. wcel 2141

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by:  spc2ev  2826  th3q  6618  addnnnq0  7411  mulnnnq0  7412  addsrpr  7707  mulsrpr  7708