Theorem cbvab

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Usage of the weaker cbvabw and cbvabv are preferred. (Contributed by Andrew Salmon, 11-Jul-2011) (Proof shortened by Wolf Lammen, 16-Nov-2019) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbvab.1	|- F/ y ph
2		cbvab.2	|- F/ x ps
3		cbvab.3	|- ( x = y -> ( ph <-> ps ) )
4	1	sbco2	|- ( [ z / y ] [ y / x ] ph <-> [ z / x ] ph )
5	2 3	sbie	|- ( [ y / x ] ph <-> ps )
6	5	sbbii	|- ( [ z / y ] [ y / x ] ph <-> [ z / y ] ps )
7	4 6	bitr3i	|- ( [ z / x ] ph <-> [ z / y ] ps )
8		df-clab	|- ( z e. { x | ph } <-> [ z / x ] ph )
9		df-clab	|- ( z e. { y | ps } <-> [ z / y ] ps )
10	7 8 9	3bitr4i	|- ( z e. { x | ph } <-> z e. { y | ps } )
11	10	eqriv	|- { x | ph } = { y | ps }