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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 ) )