Theorem sepexi

Description: Convert implication to equivalence within an existence statement using the Separation Scheme (Aussonderung) ax-sep . Inference associated with sepex . (Contributed by NM, 21-Jun-1993) Generalize conclusion, extract closed form, and avoid ax-9 . (Revised by Matthew House, 19-Sep-2025)

	Ref	Expression
Hypothesis	sepexi.1	|- E. y A. x ( ph -> x e. y )
Assertion	sepexi	|- E. z A. x ( x e. z <-> ph )

Proof

Step	Hyp	Ref	Expression
1		sepexi.1	|- E. y A. x ( ph -> x e. y )
2		sepex	|- ( E. y A. x ( ph -> x e. y ) -> E. z A. x ( x e. z <-> ph ) )
3	1 2	ax-mp	|- E. z A. x ( x e. z <-> ph )