Theorem ralimdvva 2540

Description: Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1451). (Contributed by AV, 27-Nov-2019.)

Hypothesis

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

Assertion

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

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

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

Proof of Theorem ralimdvva

Step	Hyp	Ref	Expression
1		ralimdvva.1	. . . 4 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps -> ch ) )
2	1	anassrs 398	. . 3 |- ( ( ( ph /\ x e. A ) /\ y e. B ) -> ( ps -> ch ) )
3	2	ralimdva 2538	. 2 |- ( ( ph /\ x e. A ) -> ( A. y e. B ps -> A. y e. B ch ) )
4	3	ralimdva 2538	1 |- ( ph -> ( A. x e. A A. y e. B ps -> A. x e. A A. y e. B ch ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2142   A.wral 2449

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 1441  ax-gen 1443  ax-4 1504  ax-17 1520

This theorem depends on definitions:  df-bi 116  df-nf 1455  df-ral 2454

This theorem is referenced by:  addcncntoplem  13430