Metamath Proof Explorer


Theorem reximdvai

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of Margaris p. 90. (Contributed by NM, 14-Nov-2002) Reduce dependencies on axioms. (Revised by Wolf Lammen, 8-Jan-2020)

Ref Expression
Hypothesis reximdvai.1
|- ( ph -> ( x e. A -> ( ps -> ch ) ) )
Assertion reximdvai
|- ( ph -> ( E. x e. A ps -> E. x e. A ch ) )

Proof

Step Hyp Ref Expression
1 reximdvai.1
 |-  ( ph -> ( x e. A -> ( ps -> ch ) ) )
2 1 ralrimiv
 |-  ( ph -> A. x e. A ( ps -> ch ) )
3 rexim
 |-  ( A. x e. A ( ps -> ch ) -> ( E. x e. A ps -> E. x e. A ch ) )
4 2 3 syl
 |-  ( ph -> ( E. x e. A ps -> E. x e. A ch ) )