Theorem nfrab

Description: A variable not free in a wff remains so in a restricted class abstraction. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker nfrabw when possible. (Contributed by NM, 13-Oct-2003) (Revised by Mario Carneiro, 9-Oct-2016) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		nfrab.1	|- F/ x ph
2		nfrab.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 z e. A
6		eleq1w	|- ( z = y -> ( z e. A <-> y e. A ) )
7	5 6	dvelimnf	|- ( -. A. x x = y -> F/ x y e. A )
8	1	a1i	|- ( -. A. x x = y -> F/ x ph )
9	7 8	nfand	|- ( -. A. x x = y -> F/ x ( y e. A /\ ph ) )
10	9	adantl	|- ( ( T. /\ -. A. x x = y ) -> F/ x ( y e. A /\ ph ) )
11	4 10	nfabd2	|- ( T. -> F/_ x { y | ( y e. A /\ ph ) } )
12	11	mptru	|- F/_ x { y | ( y e. A /\ ph ) }
13	3 12	nfcxfr	|- F/_ x { y e. A | ph }