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Metamath Proof Explorer

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) Avoid ax-13 . (Revised by GG, 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 }