Theorem ralrimivv 2705

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	exp3a 425	. . 3 |- (ph -> (x e. A -> (y e. B -> ps)))
3	2	ralrimdv 2703	. 2 |- (ph -> (x e. A -> A.y e. B ps))
4	3	ralrimiv 2696	1 |- (ph -> A.x e. A A.y e. B ps)

Syntax hints:   -> wi 4   /\ wa 358   e. wcel 1710  A.wral 2614

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2619

This theorem is referenced by:  ralrimivva  2706  ralrimdvv  2708  reuind  3039  nnpweq  4523  trrd  5925  connexrd  5930