Theorem ceqsralv

Description: Restricted quantifier version of ceqsalv . (Contributed by NM, 21-Jun-2013) Avoid ax-9 , ax-12 , ax-ext . (Revised by SN, 8-Sep-2024)

	Ref	Expression
Hypothesis	ceqsralv.2	|- ( x = A -> ( ph <-> ps ) )
Assertion	ceqsralv	|- ( A e. B -> ( A. x e. B ( x = A -> ph ) <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		ceqsralv.2	|- ( x = A -> ( ph <-> ps ) )
2	1	pm5.74i	|- ( ( x = A -> ph ) <-> ( x = A -> ps ) )
3	2	ralbii	|- ( A. x e. B ( x = A -> ph ) <-> A. x e. B ( x = A -> ps ) )
4		r19.23v	|- ( A. x e. B ( x = A -> ps ) <-> ( E. x e. B x = A -> ps ) )
5		risset	|- ( A e. B <-> E. x e. B x = A )
6		pm5.5	|- ( E. x e. B x = A -> ( ( E. x e. B x = A -> ps ) <-> ps ) )
7	5 6	sylbi	|- ( A e. B -> ( ( E. x e. B x = A -> ps ) <-> ps ) )
8	4 7	bitrid	|- ( A e. B -> ( A. x e. B ( x = A -> ps ) <-> ps ) )
9	3 8	bitrid	|- ( A e. B -> ( A. x e. B ( x = A -> ph ) <-> ps ) )