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)

	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		dfin5	$$⊢A\cap B=\left\{y\in A|y\in B\right\}$$
4	2	nfcri	$$⊢Ⅎxy\in B$$
5	4 1	nfrabw	$$⊢\underset{\_}{Ⅎ}x\left\{y\in A|y\in B\right\}$$
6	3 5	nfcxfr	$$⊢\underset{\_}{Ⅎ}x\left(A\cap B\right)$$