Metamath Proof Explorer


Theorem r19.28v

Description: Restricted quantifier version of one direction of 19.28 . (Assuming F/_ x A , the other direction holds when A is nonempty, see r19.28zv .) (Contributed by NM, 2-Apr-2004) (Proof shortened by Wolf Lammen, 17-Jun-2023)

Ref Expression
Assertion r19.28v
|- ( ( ph /\ A. x e. A ps ) -> A. x e. A ( ph /\ ps ) )

Proof

Step Hyp Ref Expression
1 id
 |-  ( ph -> ph )
2 1 ralrimivw
 |-  ( ph -> A. x e. A ph )
3 2 anim1i
 |-  ( ( ph /\ A. x e. A ps ) -> ( A. x e. A ph /\ A. x e. A ps ) )
4 r19.26
 |-  ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) )
5 3 4 sylibr
 |-  ( ( ph /\ A. x e. A ps ) -> A. x e. A ( ph /\ ps ) )