Theorem rexlimdvv 2483

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 255	. . 3 |- ( ( ph /\ x e. A ) -> ( y e. B -> ( ps -> ch ) ) )
3	2	rexlimdv 2476	. 2 |- ( ( ph /\ x e. A ) -> ( E. y e. B ps -> ch ) )
4	3	rexlimdva 2477	1 |- ( ph -> ( E. x e. A E. y e. B ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1433   E.wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-ral 2353  df-rex 2354

This theorem is referenced by:  rexlimdvva  2484  f1oiso2  5486  xpdom2  6328  genpcdl  6709  genpcuu  6710  distrlem1prl  6772  distrlem1pru  6773  distrlem5prl  6776  distrlem5pru  6777  recexprlemss1l  6825  recexprlemss1u  6826  qaddcl  8720  qmulcl  8722  dvdsgcd  10401  gcddiv  10408