Theorem spc2egv 3295

Description: Existential specialization with two 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 3215	. . . 4 |- ( A e. V -> E. x x = A )
2		elisset 3215	. . . 4 |- ( B e. W -> E. y y = B )
3	1, 2	anim12i 590	. . 3 |- ( ( A e. V /\ B e. W ) -> ( E. x x = A /\ E. y y = B ) )
4		eeanv 2182	. . 3 |- ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
5	3, 4	sylibr 224	. 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 240	. . 3 |- ( ps -> ( ( x = A /\ y = B ) -> ph ) )
8	7	2eximdv 1848	. 2 |- ( ps -> ( E. x E. y ( x = A /\ y = B ) -> E. x E. y ph ) )
9	5, 8	syl5com 31	1 |- ( ( A e. V /\ B e. W ) -> ( ps -> E. x E. y ph ) )

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:  spc2gv  3296  spc2ev  3301  tpres  6466  addsrpr  9896  mulsrpr  9897  2pthon3v  26839  umgr2wlk  26845  0pthonv  26990  1pthon2v  27013  dvnprodlem1  40161