Theorem ceqsalv

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993) Avoid ax-12 . (Revised by SN, 8-Sep-2024)

	Ref	Expression
Hypotheses	ceqsalv.1	|- A e. _V
	ceqsalv.2	|- ( x = A -> ( ph <-> ps ) )
Assertion	ceqsalv	|- ( A. x ( x = A -> ph ) <-> ps )

Proof

Step	Hyp	Ref	Expression
1		ceqsalv.1	|- A e. _V
2		ceqsalv.2	|- ( x = A -> ( ph <-> ps ) )
3	2	pm5.74i	|- ( ( x = A -> ph ) <-> ( x = A -> ps ) )
4	3	albii	|- ( A. x ( x = A -> ph ) <-> A. x ( x = A -> ps ) )
5		19.23v	|- ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) )
6	1	isseti	|- E. x x = A
7		pm5.5	|- ( E. x x = A -> ( ( E. x x = A -> ps ) <-> ps ) )
8	6 7	ax-mp	|- ( ( E. x x = A -> ps ) <-> ps )
9	4 5 8	3bitri	|- ( A. x ( x = A -> ph ) <-> ps )