Metamath Proof Explorer


Theorem ceqsal

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 df-clab . (Revised by Wolf Lammen, 23-Jan-2025)

Ref Expression
Hypotheses ceqsal.1
|- F/ x ps
ceqsal.2
|- A e. _V
ceqsal.3
|- ( x = A -> ( ph <-> ps ) )
Assertion ceqsal
|- ( A. x ( x = A -> ph ) <-> ps )

Proof

Step Hyp Ref Expression
1 ceqsal.1
 |-  F/ x ps
2 ceqsal.2
 |-  A e. _V
3 ceqsal.3
 |-  ( x = A -> ( ph <-> ps ) )
4 1 19.23
 |-  ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) )
5 3 pm5.74i
 |-  ( ( x = A -> ph ) <-> ( x = A -> ps ) )
6 5 albii
 |-  ( A. x ( x = A -> ph ) <-> A. x ( x = A -> ps ) )
7 2 isseti
 |-  E. x x = A
8 7 a1bi
 |-  ( ps <-> ( E. x x = A -> ps ) )
9 4 6 8 3bitr4i
 |-  ( A. x ( x = A -> ph ) <-> ps )