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Metamath Proof Explorer

Theorem r19.36v

Description: Restricted quantifier version of one direction of 19.36 . (The other direction holds iff A is nonempty, see r19.36zv .) (Contributed by NM, 22-Oct-2003)

		Ref	Expression
	Assertion	r19.36v	\|- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> ps ) )

Proof

Step	Hyp	Ref	Expression
1		r19.35	\|- ( E. x e. A ( ph -> ps ) <-> ( A. x e. A ph -> E. x e. A ps ) )
2		id	\|- ( ps -> ps )
3	2	rexlimivw	\|- ( E. x e. A ps -> ps )
4	3	imim2i	\|- ( ( A. x e. A ph -> E. x e. A ps ) -> ( A. x e. A ph -> ps ) )
5	1 4	sylbi	\|- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> ps ) )