Theorem ralrimdvv 2709

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 1-Jun-2005.)

Hypothesis

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

Assertion

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

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

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

Proof of Theorem ralrimdvv

Step	Hyp	Ref	Expression
1		ralrimdvv.1	. . . 4 |- (ph -> (ps -> ((x e. A /\ y e. B) -> ch)))
2	1	imp 418	. . 3 |- ((ph /\ ps) -> ((x e. A /\ y e. B) -> ch))
3	2	ralrimivv 2706	. 2 |- ((ph /\ ps) -> A.x e. A A.y e. B ch)
4	3	ex 423	1 |- (ph -> (ps -> A.x e. A A.y e. B ch))

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

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-11 1746

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

This theorem is referenced by:  ralrimdvva  2710