Theorem ralrimivv 2575

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 2573	. 2 |- ( ph -> ( x e. A -> A. y e. B ps ) )
4	3	ralrimiv 2566	1 |- ( ph -> A. x e. A A. y e. B ps )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2164   A.wral 2472

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 1458  ax-gen 1460  ax-4 1521  ax-17 1537

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-ral 2477

This theorem is referenced by:  ralrimivva  2576  ralrimdvv  2578  reuind  2966  ssrel2  4750  f1o2ndf1  6283  smoiso  6357  nndifsnid  6562  receuap  8690  lbreu  8966  0subm  13059  insubm  13060  iscmnd  13371  quscrng  14032  tgcl  14243  topbas  14246  epttop  14269  restbasg  14347  txbas  14437  txbasval  14446  blfps  14588  blf  14589  blbas  14612