Theorem bnj1441

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) Add disjoint variable condition to avoid ax-13 . See bnj1441g for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1441.1	|- ( x e. A -> A. y x e. A )
	bnj1441.2	|- ( ph -> A. y ph )
Assertion	bnj1441	|- ( z e. { x e. A | ph } -> A. y z e. { x e. A | ph } )

Proof

Step	Hyp	Ref	Expression
1		bnj1441.1	|- ( x e. A -> A. y x e. A )
2		bnj1441.2	|- ( ph -> A. y ph )
3		df-rab	|- { x e. A | ph } = { x | ( x e. A /\ ph ) }
4	1 2	hban	|- ( ( x e. A /\ ph ) -> A. y ( x e. A /\ ph ) )
5	4	hbab	|- ( z e. { x | ( x e. A /\ ph ) } -> A. y z e. { x | ( x e. A /\ ph ) } )
6	3 5	hbxfreq	|- ( z e. { x e. A | ph } -> A. y z e. { x e. A | ph } )