Theorem rabbi

Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbii . (Contributed by NM, 25-Nov-2013)

	Ref	Expression
Assertion	rabbi	|- ( A. x e. A ( ps <-> ch ) <-> { x e. A | ps } = { x e. A | ch } )

Proof

Step	Hyp	Ref	Expression
1		abbib	|- ( { x | ( x e. A /\ ps ) } = { x | ( x e. A /\ ch ) } <-> A. x ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
2		df-rab	|- { x e. A | ps } = { x | ( x e. A /\ ps ) }
3		df-rab	|- { x e. A | ch } = { x | ( x e. A /\ ch ) }
4	2 3	eqeq12i	|- ( { x e. A | ps } = { x e. A | ch } <-> { x | ( x e. A /\ ps ) } = { x | ( x e. A /\ ch ) } )
5		df-ral	|- ( A. x e. A ( ps <-> ch ) <-> A. x ( x e. A -> ( ps <-> ch ) ) )
6		pm5.32	|- ( ( x e. A -> ( ps <-> ch ) ) <-> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
7	6	albii	|- ( A. x ( x e. A -> ( ps <-> ch ) ) <-> A. x ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
8	5 7	bitri	|- ( A. x e. A ( ps <-> ch ) <-> A. x ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
9	1 4 8	3bitr4ri	|- ( A. x e. A ( ps <-> ch ) <-> { x e. A | ps } = { x e. A | ch } )