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Metamath Proof Explorer

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 GG, 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	cbvsbvf	\|- ( [ 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 }