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 ) )