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 ) )