Theorem rexlimdvv 2594

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 257	. . 3 |- ( ( ph /\ x e. A ) -> ( y e. B -> ( ps -> ch ) ) )
3	2	rexlimdv 2586	. 2 |- ( ( ph /\ x e. A ) -> ( E. y e. B ps -> ch ) )
4	3	rexlimdva 2587	1 |- ( ph -> ( E. x e. A E. y e. B ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   E.wrex 2449

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-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-ral 2453  df-rex 2454

This theorem is referenced by:  rexlimdvva  2595  f1oiso2  5803  xpdom2  6805  genpcdl  7468  genpcuu  7469  distrlem1prl  7531  distrlem1pru  7532  distrlem5prl  7535  distrlem5pru  7536  recexprlemss1l  7584  recexprlemss1u  7585  qaddcl  9581  qmulcl  9583  summodc  11333  dvdsgcd  11954  gcddiv  11961  pceu  12236  pcqcl  12247  txcnp  12986  blssps  13142  blss  13143  tgqioo  13262