Metamath Proof Explorer


Theorem r19.27v

Description: Restricted quantitifer version of one direction of 19.27 . (Assuming F/_ x A , the other direction holds when A is nonempty, see r19.27zv .) (Contributed by NM, 3-Jun-2004) (Proof shortened by Andrew Salmon, 30-May-2011) (Proof shortened by Wolf Lammen, 17-Jun-2023)

Ref Expression
Assertion r19.27v
|- ( ( A. x e. A ph /\ ps ) -> A. x e. A ( ph /\ ps ) )

Proof

Step Hyp Ref Expression
1 id
 |-  ( ps -> ps )
2 1 ralrimivw
 |-  ( ps -> A. x e. A ps )
3 2 anim2i
 |-  ( ( A. x e. A ph /\ 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
 |-  ( ( A. x e. A ph /\ ps ) -> A. x e. A ( ph /\ ps ) )