Theorem abn0

Description: Nonempty class abstraction. See also ab0 . (Contributed by NM, 26-Dec-1996) (Proof shortened by Mario Carneiro, 11-Nov-2016) Avoid df-clel , ax-8 . (Revised by Gino Giotto, 30-Aug-2024)

	Ref	Expression
Assertion	abn0	|- ( { x | ph } =/= (/) <-> E. x ph )

Proof

Step	Hyp	Ref	Expression
1		ab0	|- ( { x | ph } = (/) <-> A. x -. ph )
2	1	notbii	|- ( -. { x | ph } = (/) <-> -. A. x -. ph )
3		df-ne	|- ( { x | ph } =/= (/) <-> -. { x | ph } = (/) )
4		df-ex	|- ( E. x ph <-> -. A. x -. ph )
5	2 3 4	3bitr4i	|- ( { x | ph } =/= (/) <-> E. x ph )