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Metamath Proof Explorer

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 ) )