Theorem rexlimdvv 2675

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 2667	. 2 |- ( ( ph /\ x e. A ) -> ( E. y e. B ps -> ch ) )
4	3	rexlimdva 2668	1 |- ( ph -> ( E. x e. A E. y e. B ps -> ch ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209   E.wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-i5r 1588

This proof depends on definitions:  df-bi 117  df-nf 1514  df-ral 2533  df-rex 2534

This theorem is used by:  rexlimdvva  2676  f1oiso2  6036  rex2dom  7116  xpdom2  7135  genpcdl  7892  genpcuu  7893  distrlem1prl  7955  distrlem1pru  7956  distrlem5prl  7959  distrlem5pru  7960  recexprlemss1l  8008  recexprlemss1u  8009  qaddcl  10050  qmulcl  10052  summodc  12174  dvdsgcd  12813  gcddiv  12820  pceu  13102  pcqcl  13113  txcnp  15514  blssps  15670  blss  15671  tgqioo  15798  upgredg2vtx  16611