Theorem r19.23

Description: Restricted quantifier version of 19.23 . See r19.23v for a version requiring fewer axioms. (Contributed by NM, 22-Oct-2010) (Proof shortened by Mario Carneiro, 8-Oct-2016)

	Ref	Expression
Hypothesis	r19.23.1	|- F/ x ps
Assertion	r19.23	|- ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) )

Proof

Step	Hyp	Ref	Expression
1		r19.23.1	|- F/ x ps
2		r19.23t	|- ( F/ x ps -> ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) ) )
3	1 2	ax-mp	|- ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) )