Theorem abbiOLD

Description: Obsolete proof of abbi as of 7-Jan-2024. (Contributed by NM, 25-Nov-2013) (Revised by Mario Carneiro, 11-Aug-2016) (Proof shortened by Wolf Lammen, 16-Nov-2019) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		hbab1	|- ( y e. { x | ph } -> A. x y e. { x | ph } )
2		hbab1	|- ( y e. { x | ps } -> A. x y e. { x | ps } )
3	1 2	cleqh	|- ( { x | ph } = { x | ps } <-> A. x ( x e. { x | ph } <-> x e. { x | ps } ) )
4		abid	|- ( x e. { x | ph } <-> ph )
5		abid	|- ( x e. { x | ps } <-> ps )
6	4 5	bibi12i	|- ( ( x e. { x | ph } <-> x e. { x | ps } ) <-> ( ph <-> ps ) )
7	6	albii	|- ( A. x ( x e. { x | ph } <-> x e. { x | ps } ) <-> A. x ( ph <-> ps ) )
8	3 7	bitr2i	|- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } )