Theorem cbvsbcw

Description: Change bound variables in a wff substitution. Version of cbvsbc with a disjoint variable condition, which does not require ax-13 . (Contributed by Jeff Hankins, 19-Sep-2009) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypotheses	cbvsbcw.1	|- F/ y ph
	cbvsbcw.2	|- F/ x ps
	cbvsbcw.3	|- ( x = y -> ( ph <-> ps ) )
Assertion	cbvsbcw	|- ( [. A / x ]. ph <-> [. A / y ]. ps )

Proof

Step	Hyp	Ref	Expression
1		cbvsbcw.1	|- F/ y ph
2		cbvsbcw.2	|- F/ x ps
3		cbvsbcw.3	|- ( x = y -> ( ph <-> ps ) )
4	1 2 3	cbvabw	|- { x | ph } = { y | ps }
5	4	eleq2i	|- ( A e. { x | ph } <-> A e. { y | ps } )
6		df-sbc	|- ( [. A / x ]. ph <-> A e. { x | ph } )
7		df-sbc	|- ( [. A / y ]. ps <-> A e. { y | ps } )
8	5 6 7	3bitr4i	|- ( [. A / x ]. ph <-> [. A / y ]. ps )