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 ) )