Theorem nfrabw

Description: A variable not free in a wff remains so in a restricted class abstraction. Version of nfrab with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 13-Oct-2003) (Revised by Gino Giotto, 10-Jan-2024) (Proof shortened by Wolf Lammen, 23-Nov-2024)

	Ref	Expression
Hypotheses	nfrabw.1	|- F/ x ph
	nfrabw.2	|- F/_ x A
Assertion	nfrabw	|- F/_ x { y e. A | ph }

Proof

Step	Hyp	Ref	Expression
1		nfrabw.1	|- F/ x ph
2		nfrabw.2	|- F/_ x A
3		df-rab	|- { y e. A | ph } = { y | ( y e. A /\ ph ) }
4		nftru	|- F/ y T.
5	2	nfcri	|- F/ x y e. A
6	5 1	nfan	|- F/ x ( y e. A /\ ph )
7	6	a1i	|- ( T. -> F/ x ( y e. A /\ ph ) )
8	4 7	nfabdw	|- ( T. -> F/_ x { y | ( y e. A /\ ph ) } )
9	8	mptru	|- F/_ x { y | ( y e. A /\ ph ) }
10	3 9	nfcxfr	|- F/_ x { y e. A | ph }