Theorem ralrimivv 2578

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 2576	. 2 |- ( ph -> ( x e. A -> A. y e. B ps ) )
4	3	ralrimiv 2569	1 |- ( ph -> A. x e. A A. y e. B ps )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2167   A.wral 2475

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 1461  ax-gen 1463  ax-4 1524  ax-17 1540

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-ral 2480

This theorem is referenced by:  ralrimivva  2579  ralrimdvv  2581  reuind  2969  ssrel2  4754  f1o2ndf1  6287  smoiso  6361  nndifsnid  6566  receuap  8698  lbreu  8974  0subm  13126  insubm  13127  iscmnd  13438  quscrng  14099  tgcl  14310  topbas  14313  epttop  14336  restbasg  14414  txbas  14504  txbasval  14513  blfps  14655  blf  14656  blbas  14679