Theorem abbib

Description: Equal class abstractions require equivalent formulas, and conversely. (Contributed by NM, 25-Nov-2013) (Revised by Mario Carneiro, 11-Aug-2016) Remove dependency on ax-8 and df-clel (by avoiding use of cleqh ). (Revised by BJ, 23-Jun-2019) Definitial form. (Revised by Wolf Lammen, 23-Feb-2025)

	Ref	Expression
Assertion	abbib	|- ( { x | ph } = { x | ps } <-> A. x ( ph <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		dfcleq	|- ( { x | ph } = { x | ps } <-> A. y ( y e. { x | ph } <-> y e. { x | ps } ) )
2		nfsab1	|- F/ x y e. { x | ph }
3		nfsab1	|- F/ x y e. { x | ps }
4	2 3	nfbi	|- F/ x ( y e. { x | ph } <-> y e. { x | ps } )
5		nfv	|- F/ y ( ph <-> ps )
6		df-clab	|- ( y e. { x | ph } <-> [ y / x ] ph )
7		sbequ12r	|- ( y = x -> ( [ y / x ] ph <-> ph ) )
8	6 7	bitrid	|- ( y = x -> ( y e. { x | ph } <-> ph ) )
9		df-clab	|- ( y e. { x | ps } <-> [ y / x ] ps )
10		sbequ12r	|- ( y = x -> ( [ y / x ] ps <-> ps ) )
11	9 10	bitrid	|- ( y = x -> ( y e. { x | ps } <-> ps ) )
12	8 11	bibi12d	|- ( y = x -> ( ( y e. { x | ph } <-> y e. { x | ps } ) <-> ( ph <-> ps ) ) )
13	4 5 12	cbvalv1	|- ( A. y ( y e. { x | ph } <-> y e. { x | ps } ) <-> A. x ( ph <-> ps ) )
14	1 13	bitri	|- ( { x | ph } = { x | ps } <-> A. x ( ph <-> ps ) )