Theorem ceqsal1t

Description: One direction of ceqsalt is based on fewer assumptions and fewer axioms. It is at the same time the reverse direction of vtoclgft . Extracted from a proof of ceqsalt . (Contributed by Wolf Lammen, 25-Mar-2025)

	Ref	Expression
Assertion	ceqsal1t	|- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( ps -> A. x ( x = A -> ph ) ) )

Proof

Step	Hyp	Ref	Expression
1		biimpr	|- ( ( ph <-> ps ) -> ( ps -> ph ) )
2	1	imim2i	|- ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ps -> ph ) ) )
3	2	com23	|- ( ( x = A -> ( ph <-> ps ) ) -> ( ps -> ( x = A -> ph ) ) )
4	3	alimi	|- ( A. x ( x = A -> ( ph <-> ps ) ) -> A. x ( ps -> ( x = A -> ph ) ) )
5		19.21t	|- ( F/ x ps -> ( A. x ( ps -> ( x = A -> ph ) ) <-> ( ps -> A. x ( x = A -> ph ) ) ) )
6	4 5	imbitrid	|- ( F/ x ps -> ( A. x ( x = A -> ( ph <-> ps ) ) -> ( ps -> A. x ( x = A -> ph ) ) ) )
7	6	imp	|- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) ) -> ( ps -> A. x ( x = A -> ph ) ) )