Theorem ralrimdvva 2974

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 2-Feb-2008.)

Hypothesis

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

Assertion

Ref	Expression
ralrimdvva	|- ( 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 ralrimdvva

Step	Hyp	Ref	Expression
1		ralrimdvva.1	. . . 4 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps -> ch ) )
2	1	ex 450	. . 3 |- ( ph -> ( ( x e. A /\ y e. B ) -> ( ps -> ch ) ) )
3	2	com23 86	. 2 |- ( ph -> ( ps -> ( ( x e. A /\ y e. B ) -> ch ) ) )
4	3	ralrimdvv 2973	1 |- ( ph -> ( ps -> A. x e. A A. y e. B ch ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839

This theorem depends on definitions:  df-bi 197  df-an 386  df-ral 2917

This theorem is referenced by:  isosolem  6597  kgencn2  21360  fbunfip  21673  reconn  22631  c1lip1  23760  cdj3i  29300  poimirlem29  33438  ispridl2  33837  ispridlc  33869