Theorem ralrimdvv 2973

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 445	. . 3 |- ( ( ph /\ ps ) -> ( ( x e. A /\ y e. B ) -> ch ) )
3	2	ralrimivv 2970	. 2 |- ( ( ph /\ ps ) -> A. x e. A A. y e. B ch )
4	3	ex 450	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:  ralrimdvva  2974  lspsneu  19123  pmatcoe1fsupp  20506  aalioulem4  24090  fargshiftf1  41377