Theorem rexlimdvv 2655

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 2647	. 2 |- ( ( ph /\ x e. A ) -> ( E. y e. B ps -> ch ) )
4	3	rexlimdva 2648	1 |- ( ph -> ( E. x e. A E. y e. B ps -> ch ) )

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:  rexlimdvva  2656  f1oiso2  5963  rex2dom  6991  xpdom2  7010  genpcdl  7729  genpcuu  7730  distrlem1prl  7792  distrlem1pru  7793  distrlem5prl  7796  distrlem5pru  7797  recexprlemss1l  7845  recexprlemss1u  7846  qaddcl  9859  qmulcl  9861  summodc  11934  dvdsgcd  12573  gcddiv  12580  pceu  12858  pcqcl  12869  txcnp  14985  blssps  15141  blss  15142  tgqioo  15269  upgredg2vtx  15987