Theorem ralrimivv 2611

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 24-Jul-2004.)

Hypothesis

Ref	Expression
ralrimivv.1	|- ( ph -> ( ( x e. A /\ y e. B ) -> ps ) )

Assertion

Ref	Expression
ralrimivv	|- ( ph -> A. x e. A A. y e. B ps )

Distinct variable groups:   x, y, ph   y, A

Allowed substitution hints:   ps( x, y)   A( x)   B( x, y)

Proof of Theorem ralrimivv

Step	Hyp	Ref	Expression
1		ralrimivv.1	. . . 4 |- ( ph -> ( ( x e. A /\ y e. B ) -> ps ) )
2	1	expd 258	. . 3 |- ( ph -> ( x e. A -> ( y e. B -> ps ) ) )
3	2	ralrimdv 2609	. 2 |- ( ph -> ( x e. A -> A. y e. B ps ) )
4	3	ralrimiv 2602	1 |- ( ph -> A. x e. A A. y e. B ps )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   A.wral 2508

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-4 1556  ax-17 1572

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-ral 2513

This theorem is referenced by:  ralrimivva  2612  ralrimdvv  2614  reuind  3009  ssrel2  4814  f1o2ndf1  6388  smoiso  6463  nndifsnid  6670  receuap  8839  lbreu  9115  0subm  13557  insubm  13558  iscmnd  13875  quscrng  14537  tgcl  14778  topbas  14781  epttop  14804  restbasg  14882  txbas  14972  txbasval  14981  blfps  15123  blf  15124  blbas  15147