Metamath Proof Explorer


Theorem reximdvva

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

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

Proof

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