Theorem cbvsbv

Description: Change the bound variable (i.e. the substituted one) in wff's linked by implicit substitution. The proof was extracted from a former cbvabv version. (Contributed by Wolf Lammen, 16-Mar-2025)

	Ref	Expression
Hypothesis	cbvsbv.1	|- ( x = y -> ( ph <-> ps ) )
Assertion	cbvsbv	|- ( [ z / x ] ph <-> [ z / y ] ps )

Proof

Step	Hyp	Ref	Expression
1		cbvsbv.1	|- ( x = y -> ( ph <-> ps ) )
2		sbco2vv	|- ( [ z / y ] [ y / x ] ph <-> [ z / x ] ph )
3	1	sbievw	|- ( [ y / x ] ph <-> ps )
4	3	sbbii	|- ( [ z / y ] [ y / x ] ph <-> [ z / y ] ps )
5	2 4	bitr3i	|- ( [ z / x ] ph <-> [ z / y ] ps )