Theorem eqabcbw

Description: Version of eqabcb using implicit substitution, which requires fewer axioms. (Contributed by TM, 24-Jan-2026)

	Ref	Expression
Hypothesis	eqabbw.1	|- ( x = y -> ( ph <-> ps ) )
Assertion	eqabcbw	|- ( { x | ph } = A <-> A. y ( ps <-> y e. A ) )

Proof

Step	Hyp	Ref	Expression
1		eqabbw.1	|- ( x = y -> ( ph <-> ps ) )
2	1	eqabbw	|- ( A = { x | ph } <-> A. y ( y e. A <-> ps ) )
3		eqcom	|- ( { x | ph } = A <-> A = { x | ph } )
4		bicom	|- ( ( ps <-> y e. A ) <-> ( y e. A <-> ps ) )
5	4	albii	|- ( A. y ( ps <-> y e. A ) <-> A. y ( y e. A <-> ps ) )
6	2 3 5	3bitr4i	|- ( { x | ph } = A <-> A. y ( ps <-> y e. A ) )