Theorem eqabdv

Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994) Avoid ax-11 . (Revised by Wolf Lammen, 6-May-2023)

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

Proof

Step	Hyp	Ref	Expression
1		eqabdv.1	|- ( ph -> ( x e. A <-> ps ) )
2	1	sbbidv	|- ( ph -> ( [ y / x ] x e. A <-> [ y / x ] ps ) )
3		clelsb1	|- ( [ y / x ] x e. A <-> y e. A )
4	3	bicomi	|- ( y e. A <-> [ y / x ] x e. A )
5		df-clab	|- ( y e. { x | ps } <-> [ y / x ] ps )
6	2 4 5	3bitr4g	|- ( ph -> ( y e. A <-> y e. { x | ps } ) )
7	6	eqrdv	|- ( ph -> A = { x | ps } )