Theorem ralrimivv 2400

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 245	. . 3 |- ( ph -> ( x e. A -> ( y e. B -> ps ) ) )
3	2	ralrimdv 2398	. 2 |- ( ph -> ( x e. A -> A. y e. B ps ) )
4	3	ralrimiv 2391	1 |- ( ph -> A. x e. A A. y e. B ps )

Syntax hints:   -> wi 4   /\ wa 97   e. wcel 1393   A.wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311

This theorem is referenced by:  ralrimivva  2401  ralrimdvv  2403  reuind  2744  ssrel2  4430  f1o2ndf1  5849  smoiso  5917  receuap  7650