Metamath Proof Explorer


Theorem ralrimdva

Description: Inference from Theorem 19.21 of Margaris p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Feb-2008) (Proof shortened by Wolf Lammen, 28-Dec-2019)

Ref Expression
Hypothesis ralrimdva.1
|- ( ( ph /\ x e. A ) -> ( ps -> ch ) )
Assertion ralrimdva
|- ( ph -> ( ps -> A. x e. A ch ) )

Proof

Step Hyp Ref Expression
1 ralrimdva.1
 |-  ( ( ph /\ x e. A ) -> ( ps -> ch ) )
2 1 expimpd
 |-  ( ph -> ( ( x e. A /\ ps ) -> ch ) )
3 2 expcomd
 |-  ( ph -> ( ps -> ( x e. A -> ch ) ) )
4 3 ralrimdv
 |-  ( ph -> ( ps -> A. x e. A ch ) )