Theorem sbceqbidf

Description: Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018)

	Ref	Expression
Hypotheses	sbceqbidf.1	|- F/ x ph
	sbceqbidf.2	|- ( ph -> A = B )
	sbceqbidf.3	|- ( ph -> ( ps <-> ch ) )
Assertion	sbceqbidf	|- ( ph -> ( [. A / x ]. ps <-> [. B / x ]. ch ) )

Proof

Step	Hyp	Ref	Expression
1		sbceqbidf.1	|- F/ x ph
2		sbceqbidf.2	|- ( ph -> A = B )
3		sbceqbidf.3	|- ( ph -> ( ps <-> ch ) )
4	1 3	abbid	|- ( ph -> { x | ps } = { x | ch } )
5	2 4	eleq12d	|- ( ph -> ( A e. { x | ps } <-> B e. { x | ch } ) )
6		df-sbc	|- ( [. A / x ]. ps <-> A e. { x | ps } )
7		df-sbc	|- ( [. B / x ]. ch <-> B e. { x | ch } )
8	5 6 7	3bitr4g	|- ( ph -> ( [. A / x ]. ps <-> [. B / x ]. ch ) )