Theorem rexlimivv 2743

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.)

Hypothesis

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

Assertion

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

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

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

Proof of Theorem rexlimivv

Step	Hyp	Ref	Expression
1		rexlimivv.1	. . 3 |- ((x e. A /\ y e. B) -> (ph -> ps))
2	1	rexlimdva 2738	. 2 |- (x e. A -> (E.y e. B ph -> ps))
3	2	rexlimiv 2732	1 |- (E.x e. A E.y e. B ph -> ps)

Syntax hints:   -> wi 4   /\ wa 358   e. wcel 1710  E.wrex 2615

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-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2619  df-rex 2620

This theorem is referenced by:  2reu5  3044  tfin11  4493  peano4nc  6150  sbth  6206  nclenc  6222  lenc  6223  letc  6231  ce2le  6233