Metamath Proof Explorer


Theorem cbvaldva

Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvaldvaw if possible. (Contributed by David Moews, 1-May-2017) (New usage is discouraged.)

Ref Expression
Hypothesis cbvaldva.1
|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
Assertion cbvaldva
|- ( ph -> ( A. x ps <-> A. y ch ) )

Proof

Step Hyp Ref Expression
1 cbvaldva.1
 |-  ( ( ph /\ x = y ) -> ( ps <-> ch ) )
2 nfv
 |-  F/ y ph
3 nfvd
 |-  ( ph -> F/ y ps )
4 1 ex
 |-  ( ph -> ( x = y -> ( ps <-> ch ) ) )
5 2 3 4 cbvald
 |-  ( ph -> ( A. x ps <-> A. y ch ) )