Theorem nfiin

Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014) Add disjoint variable condition to avoid ax-13 . See nfiing for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024)

	Ref	Expression
Hypotheses	nfiun.1	|- F/_ y A
	nfiun.2	|- F/_ y B
Assertion	nfiin	|- F/_ y |^|_ x e. A B

Proof

Step	Hyp	Ref	Expression
1		nfiun.1	|- F/_ y A
2		nfiun.2	|- F/_ y B
3		df-iin	|- |^|_ x e. A B = { z | A. x e. A z e. B }
4	2	nfcri	|- F/ y z e. B
5	1 4	nfralw	|- F/ y A. x e. A z e. B
6	5	nfab	|- F/_ y { z | A. x e. A z e. B }
7	3 6	nfcxfr	|- F/_ y |^|_ x e. A B