Theorem nfdif

Description: Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003) (Revised by Mario Carneiro, 13-Oct-2016) Avoid ax-10 , ax-11 , ax-12 . (Revised by SN, 14-May-2025)

	Ref	Expression
Hypotheses	nfdif.1	|- F/_ x A
	nfdif.2	|- F/_ x B
Assertion	nfdif	|- F/_ x ( A \ B )

Proof

Step	Hyp	Ref	Expression
1		nfdif.1	|- F/_ x A
2		nfdif.2	|- F/_ x B
3		eldif	|- ( y e. ( A \ B ) <-> ( y e. A /\ -. y e. B ) )
4	1	nfcri	|- F/ x y e. A
5	2	nfcri	|- F/ x y e. B
6	5	nfn	|- F/ x -. y e. B
7	4 6	nfan	|- F/ x ( y e. A /\ -. y e. B )
8	3 7	nfxfr	|- F/ x y e. ( A \ B )
9	8	nfci	|- F/_ x ( A \ B )