Theorem nfiing

Description: Bound-variable hypothesis builder for indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 . See nfiin for a version with more disjoint variable conditions, but not requiring ax-13 . (Contributed by Mario Carneiro, 25-Jan-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfiung.1	$$⊢\underset{\_}{Ⅎ}yA$$
	nfiung.2	$$⊢\underset{\_}{Ⅎ}yB$$
Assertion	nfiing	$$⊢\underset{\_}{Ⅎ}y\bigcap _{x\in A}B$$

Proof

Step	Hyp	Ref	Expression
1		nfiung.1	$$⊢\underset{\_}{Ⅎ}yA$$
2		nfiung.2	$$⊢\underset{\_}{Ⅎ}yB$$
3		df-iin	$$⊢\bigcap _{x\in A}B=\left\{z|\forall x\in Az\in B\right\}$$
4	2	nfcri	$$⊢Ⅎyz\in B$$
5	1 4	nfral	$$⊢Ⅎy\forall x\in Az\in B$$
6	5	nfabg	$$⊢\underset{\_}{Ⅎ}y\left\{z|\forall x\in Az\in B\right\}$$
7	3 6	nfcxfr	$$⊢\underset{\_}{Ⅎ}y\bigcap _{x\in A}B$$