Theorem rabid2im

Description: One direction of rabid2 is based on fewer axioms. (Contributed by Wolf Lammen, 26-May-2025)

	Ref	Expression
Assertion	rabid2im	|- ( A. x e. A ph -> A = { x e. A | ph } )

Proof

Step	Hyp	Ref	Expression
1		pm4.71	|- ( ( x e. A -> ph ) <-> ( x e. A <-> ( x e. A /\ ph ) ) )
2	1	albii	|- ( A. x ( x e. A -> ph ) <-> A. x ( x e. A <-> ( x e. A /\ ph ) ) )
3		eqab	|- ( A. x ( x e. A <-> ( x e. A /\ ph ) ) -> A = { x | ( x e. A /\ ph ) } )
4	2 3	sylbi	|- ( A. x ( x e. A -> ph ) -> A = { x | ( x e. A /\ ph ) } )
5		df-ral	|- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
6		df-rab	|- { x e. A | ph } = { x | ( x e. A /\ ph ) }
7	6	eqeq2i	|- ( A = { x e. A | ph } <-> A = { x | ( x e. A /\ ph ) } )
8	4 5 7	3imtr4i	|- ( A. x e. A ph -> A = { x e. A | ph } )