Theorem ralrimdvv 2554

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 123	. . 3 |- ( ( ph /\ ps ) -> ( ( x e. A /\ y e. B ) -> ch ) )
3	2	ralrimivv 2551	. 2 |- ( ( ph /\ ps ) -> A. x e. A A. y e. B ch )
4	3	ex 114	1 |- ( ph -> ( ps -> A. x e. A A. y e. B ch ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   A.wral 2448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-ral 2453

This theorem is referenced by:  ralrimdvva  2555