Metamath Proof Explorer


Theorem sb2ae

Description: In the case of two successive substitutions for two always equal variables, the second substitution has no effect. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by BJ and WL, 9-Aug-2023) (New usage is discouraged.)

Ref Expression
Assertion sb2ae
|- ( A. x x = y -> ( [ u / x ] [ v / y ] ph <-> [ v / y ] ph ) )

Proof

Step Hyp Ref Expression
1 drsb1
 |-  ( A. x x = y -> ( [ u / x ] [ v / y ] ph <-> [ u / y ] [ v / y ] ph ) )
2 nfs1v
 |-  F/ y [ v / y ] ph
3 2 sbf
 |-  ( [ u / y ] [ v / y ] ph <-> [ v / y ] ph )
4 1 3 bitrdi
 |-  ( A. x x = y -> ( [ u / x ] [ v / y ] ph <-> [ v / y ] ph ) )