Theorem rexlimdvv 2601

Description: Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004.)

Hypothesis

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

Assertion

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

Distinct variable groups:   x, y, ph   ch, x, y   y, A

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

Proof of Theorem rexlimdvv

Step	Hyp	Ref	Expression
1		rexlimdvv.1	. . . 4 |- ( ph -> ( ( x e. A /\ y e. B ) -> ( ps -> ch ) ) )
2	1	expdimp 259	. . 3 |- ( ( ph /\ x e. A ) -> ( y e. B -> ( ps -> ch ) ) )
3	2	rexlimdv 2593	. 2 |- ( ( ph /\ x e. A ) -> ( E. y e. B ps -> ch ) )
4	3	rexlimdva 2594	1 |- ( ph -> ( E. x e. A E. y e. B ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2148   E.wrex 2456

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-ral 2460  df-rex 2461

This theorem is referenced by:  rexlimdvva  2602  f1oiso2  5828  xpdom2  6831  genpcdl  7518  genpcuu  7519  distrlem1prl  7581  distrlem1pru  7582  distrlem5prl  7585  distrlem5pru  7586  recexprlemss1l  7634  recexprlemss1u  7635  qaddcl  9635  qmulcl  9637  summodc  11391  dvdsgcd  12013  gcddiv  12020  pceu  12295  pcqcl  12306  txcnp  13774  blssps  13930  blss  13931  tgqioo  14050