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Metamath Proof Explorer

Theorem vtocl

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993) Remove dependency on ax-10 . (Revised by BJ, 29-Nov-2020) (Proof shortened by SN, 20-Apr-2024) (Proof shortened by Wolf Lammen, 20-Jun-2025)

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

Proof

Step Hyp Ref Expression
1 vtocl.1 
 |-  A e. _V
2 vtocl.2 
 |-  ( x = A -> ( ph <-> ps ) )
3 vtocl.3 
 |-  ph
4 3 2 mpbii 
 |-  ( x = A -> ps )
5 1 4 vtocle 
 |-  ps