Theorem rexlimdvv 2745

Description: Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004.)

Hypothesis

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

Assertion

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

Distinct variable groups:   x,y,ph   ch,x,y   y,A

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

Proof of Theorem rexlimdvv

Step	Hyp	Ref	Expression
1		rexlimdvv.1	. . . 4 |- (ph -> ((x e. A /\ y e. B) -> (ps -> ch)))
2	1	expdimp 426	. . 3 |- ((ph /\ x e. A) -> (y e. B -> (ps -> ch)))
3	2	rexlimdv 2738	. 2 |- ((ph /\ x e. A) -> (E.y e. B ps -> ch))
4	3	rexlimdva 2739	1 |- (ph -> (E.x e. A E.y e. B ps -> ch))

Syntax hints:   -> wi 4   /\ wa 358   e. wcel 1710  E.wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2620  df-rex 2621

This theorem is referenced by:  rexlimdvva  2746  ncfinraise  4482  ncfinlower  4484  nnpw1ex  4485  nnpweq  4524  sfinltfin  4536  f1oiso2  5501  addcdi  6251