Metamath Proof Explorer


Theorem cbvalv

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See cbvalvw for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 5-Aug-1993) Remove dependency on ax-10 , shorten. (Revised by Wolf Lammen, 11-Sep-2023) (New usage is discouraged.)

Ref Expression
Hypothesis cbvalv.1
|- ( x = y -> ( ph <-> ps ) )
Assertion cbvalv
|- ( A. x ph <-> A. y ps )

Proof

Step Hyp Ref Expression
1 cbvalv.1
 |-  ( x = y -> ( ph <-> ps ) )
2 nfv
 |-  F/ y ph
3 nfv
 |-  F/ x ps
4 2 3 1 cbval
 |-  ( A. x ph <-> A. y ps )