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	$$⊢\underset{\_}{Ⅎ}xA$$
	nfin.2	$$⊢\underset{\_}{Ⅎ}xB$$
Assertion	nfin	$$⊢\underset{\_}{Ⅎ}x\left(A\cap B\right)$$

Proof

Step	Hyp	Ref	Expression
1		nfin.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		nfin.2	$$⊢\underset{\_}{Ⅎ}xB$$
3		elin	$$⊢y\in \left(A\cap B\right)\leftrightarrow \left(y\in A\wedge y\in B\right)$$
4	1	nfcri	$$⊢Ⅎxy\in A$$
5	2	nfcri	$$⊢Ⅎxy\in B$$
6	4 5	nfan	$$⊢Ⅎx\left(y\in A\wedge y\in B\right)$$
7	3 6	nfxfr	$$⊢Ⅎxy\in \left(A\cap B\right)$$
8	7	nfci	$$⊢\underset{\_}{Ⅎ}x\left(A\cap B\right)$$