Theorem nfint

Description: Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997) (Proof shortened by Andrew Salmon, 12-Aug-2011)

	Ref	Expression
Hypothesis	nfint.1	$$⊢\underset{\_}{Ⅎ}xA$$
Assertion	nfint	$$⊢\underset{\_}{Ⅎ}x\bigcap A$$

Proof

Step	Hyp	Ref	Expression
1		nfint.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		dfint2	$$⊢\bigcap A=\left\{y|\forall z\in Ay\in z\right\}$$
3		nfv	$$⊢Ⅎxy\in z$$
4	1 3	nfralw	$$⊢Ⅎx\forall z\in Ay\in z$$
5	4	nfab	$$⊢\underset{\_}{Ⅎ}x\left\{y|\forall z\in Ay\in z\right\}$$
6	2 5	nfcxfr	$$⊢\underset{\_}{Ⅎ}x\bigcap A$$