Theorem sbceqbid

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

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

Proof

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