Theorem rabeq0w

Description: Condition for a restricted class abstraction to be empty. Version of rabeq0 using implicit substitution, which does not require ax-10 , ax-11 , ax-12 , but requires ax-8 . (Contributed by Gino Giotto, 30-Sep-2024)

	Ref	Expression
Hypothesis	rabeq0w.1	|- ( x = y -> ( ph <-> ps ) )
Assertion	rabeq0w	|- ( { x e. A | ph } = (/) <-> A. y e. A -. ps )

Proof

Step	Hyp	Ref	Expression
1		rabeq0w.1	|- ( x = y -> ( ph <-> ps ) )
2		eleq1w	|- ( x = y -> ( x e. A <-> y e. A ) )
3	2 1	anbi12d	|- ( x = y -> ( ( x e. A /\ ph ) <-> ( y e. A /\ ps ) ) )
4	3	ab0w	|- ( { x | ( x e. A /\ ph ) } = (/) <-> A. y -. ( y e. A /\ ps ) )
5		df-rab	|- { x e. A | ph } = { x | ( x e. A /\ ph ) }
6	5	eqeq1i	|- ( { x e. A | ph } = (/) <-> { x | ( x e. A /\ ph ) } = (/) )
7		raln	|- ( A. y e. A -. ps <-> A. y -. ( y e. A /\ ps ) )
8	4 6 7	3bitr4i	|- ( { x e. A | ph } = (/) <-> A. y e. A -. ps )