Theorem reximdvva 3019

Description: Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by AV, 5-Jan-2022.)

Hypothesis

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

Assertion

Ref	Expression
reximdvva	|- ( ph -> ( E. x e. A E. y e. B ps -> E. x e. A E. 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 reximdvva

Step	Hyp	Ref	Expression
1		reximdvva.1	. . . 4 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( ps -> ch ) )
2	1	anassrs 680	. . 3 |- ( ( ( ph /\ x e. A ) /\ y e. B ) -> ( ps -> ch ) )
3	2	reximdva 3017	. 2 |- ( ( ph /\ x e. A ) -> ( E. y e. B ps -> E. y e. B ch ) )
4	3	reximdva 3017	1 |- ( ph -> ( E. x e. A E. y e. B ps -> E. x e. A E. y e. B ch ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   E.wrex 2913

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-ex 1705  df-ral 2917  df-rex 2918

This theorem is referenced by:  lcmgcdlem  15319  lsmelval2  19085  cpmadugsum  20683  axpasch  25821  frgrwopreglem5  27185  frgrwopreglem5ALT  27186  eulerpartlemgvv  30438  cvmlift2lem10  31294  ftc1anclem6  33490