Theorem 19.38a

Description: Under a non-freeness hypothesis, the implication 19.38 can be strengthened to an equivalence. See also 19.38b . (Contributed by BJ, 3-Nov-2021) (Proof shortened by Wolf Lammen, 9-Jul-2022)

	Ref	Expression
Assertion	19.38a	|- ( F/ x ph -> ( ( E. x ph -> A. x ps ) <-> A. x ( ph -> ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		19.38	|- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )
2		id	|- ( F/ x ph -> F/ x ph )
3	2	nfrd	|- ( F/ x ph -> ( E. x ph -> A. x ph ) )
4		alim	|- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )
5	3 4	syl9	|- ( F/ x ph -> ( A. x ( ph -> ps ) -> ( E. x ph -> A. x ps ) ) )
6	1 5	impbid2	|- ( F/ x ph -> ( ( E. x ph -> A. x ps ) <-> A. x ( ph -> ps ) ) )