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  5950  rex2dom  6969  xpdom2  6986  genpcdl  7702  genpcuu  7703  distrlem1prl  7765  distrlem1pru  7766  distrlem5prl  7769  distrlem5pru  7770  recexprlemss1l  7818  recexprlemss1u  7819  qaddcl  9826  qmulcl  9828  summodc  11889  dvdsgcd  12528  gcddiv  12535  pceu  12813  pcqcl  12824  txcnp  14939  blssps  15095  blss  15096  tgqioo  15223  upgredg2vtx  15940