Theorem cbvabw

Description: Rule used to change bound variables, using implicit substitution. Version of cbvab with a disjoint variable condition, which does not require ax-10 , ax-13 . (Contributed by Andrew Salmon, 11-Jul-2011) (Revised by Gino Giotto, 23-May-2024)

	Ref	Expression
Hypotheses	cbvabw.1	|- F/ y ph
	cbvabw.2	|- F/ x ps
	cbvabw.3	|- ( x = y -> ( ph <-> ps ) )
Assertion	cbvabw	|- { x | ph } = { y | ps }

Proof

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