Theorem cbvabv

Description: Rule used to change bound variables, using implicit substitution. Version of cbvab with disjoint variable conditions requiring fewer axioms. (Contributed by NM, 26-May-1999) Require x , y be disjoint to avoid ax-11 and ax-13 . (Revised by Steven Nguyen, 4-Dec-2022)

	Ref	Expression
Hypothesis	cbvabv.1	|- ( x = y -> ( ph <-> ps ) )
Assertion	cbvabv	|- { x | ph } = { y | ps }

Proof

Step	Hyp	Ref	Expression
1		cbvabv.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 )
6		df-clab	|- ( z e. { x | ph } <-> [ z / x ] ph )
7		df-clab	|- ( z e. { y | ps } <-> [ z / y ] ps )
8	5 6 7	3bitr4i	|- ( z e. { x | ph } <-> z e. { y | ps } )
9	8	eqriv	|- { x | ph } = { y | ps }