Theorem abbid

Description: Equivalent wff's yield equal class abstractions (deduction form, with nonfreeness hypothesis). (Contributed by NM, 21-Jun-1993) (Revised by Mario Carneiro, 7-Oct-2016) Avoid ax-10 and ax-11 . (Revised by Wolf Lammen, 6-May-2023)

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

Proof

Step	Hyp	Ref	Expression
1		abbid.1	|- F/ x ph
2		abbid.2	|- ( ph -> ( ps <-> ch ) )
3	1 2	alrimi	|- ( ph -> A. x ( ps <-> ch ) )
4		abbi1	|- ( A. x ( ps <-> ch ) -> { x | ps } = { x | ch } )
5	3 4	syl	|- ( ph -> { x | ps } = { x | ch } )