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	⊢ Ⅎ 𝑥 𝐴
	nfin.2	⊢ Ⅎ 𝑥 𝐵
Assertion	nfin	⊢ Ⅎ 𝑥 ( 𝐴 ∩ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		nfin.1	⊢ Ⅎ 𝑥 𝐴
2		nfin.2	⊢ Ⅎ 𝑥 𝐵
3		elin	⊢ ( 𝑦 ∈ ( 𝐴 ∩ 𝐵 ) ↔ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) )
4	1	nfcri	⊢ Ⅎ 𝑥 𝑦 ∈ 𝐴
5	2	nfcri	⊢ Ⅎ 𝑥 𝑦 ∈ 𝐵
6	4 5	nfan	⊢ Ⅎ 𝑥 ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 )
7	3 6	nfxfr	⊢ Ⅎ 𝑥 𝑦 ∈ ( 𝐴 ∩ 𝐵 )
8	7	nfci	⊢ Ⅎ 𝑥 ( 𝐴 ∩ 𝐵 )