Theorem nfin

Description: Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003) (Revised by Mario Carneiro, 14-Oct-2016) Avoid ax-10 , ax-11 , ax-12 . (Revised by SN, 14-May-2025)

	Ref	Expression
Hypotheses	nfin.1	|- F/_ x A
	nfin.2	|- F/_ x B
Assertion	nfin	|- F/_ x ( A i^i B )

Proof

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