Theorem rexlimivv 2654

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.)

Hypothesis

Ref	Expression
rexlimivv.1	|- ( ( x e. A /\ y e. B ) -> ( ph -> ps ) )

Assertion

Ref	Expression
rexlimivv	|- ( E. x e. A E. y e. B ph -> ps )

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

Allowed substitution hints:   ph( x, y)   A( x)   B( x, y)

Proof of Theorem rexlimivv

Step	Hyp	Ref	Expression
1		rexlimivv.1	. . 3 |- ( ( x e. A /\ y e. B ) -> ( ph -> ps ) )
2	1	rexlimdva 2648	. 2 |- ( x e. A -> ( E. y e. B ph -> ps ) )
3	2	rexlimiv 2642	1 |- ( E. x e. A E. y e. B ph -> ps )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   E.wrex 2509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-ral 2513  df-rex 2514

This theorem is referenced by:  opelxp  4749  f1o2ndf1  6374  xpdom2  6990  distrlem5prl  7773  distrlem5pru  7774  mulrid  8143  cnegex  8324  recexap  8800  creur  9106  creui  9107  cju  9108  elz2  9518  qre  9820  qaddcl  9830  qnegcl  9831  qmulcl  9832  qreccl  9837  elpqb  9845  fundm2domnop0  11067  replim  11370  prodmodc  12089  odd2np1  12384  opoe  12406  omoe  12407  opeo  12408  omeo  12409  qredeu  12619  pythagtriplem1  12788  pcz  12855  4sqlem1  12911  4sqlem2  12912  4sqlem4  12915  mul4sq  12917  txuni2  14930  blssioo  15227  tgioo  15228  elply  15408  2sqlem2  15794  mul2sq  15795  2sqlem7  15800  upgredgpr  15947