This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) ) )