Theorem cbvabwOLD

Description: Obsolete version of cbvabw as of 23-May-2024. (Contributed by Andrew Salmon, 11-Jul-2011) (Revised by Gino Giotto, 10-Jan-2024) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbvabwOLD.1	|- F/ y ph
2		cbvabwOLD.2	|- F/ x ps
3		cbvabwOLD.3	|- ( x = y -> ( ph <-> ps ) )
4	1	sbco2v	|- ( [ z / y ] [ y / x ] ph <-> [ z / x ] ph )
5	2 3	sbiev	|- ( [ 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 }