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	⊢ Ⅎ 𝑦 𝐴
	nfiun.2	⊢ Ⅎ 𝑦 𝐵
Assertion	nfiin	⊢ Ⅎ 𝑦 ∩ 𝑥 ∈ 𝐴 𝐵

Proof

Step	Hyp	Ref	Expression
1		nfiun.1	⊢ Ⅎ 𝑦 𝐴
2		nfiun.2	⊢ Ⅎ 𝑦 𝐵
3		df-iin	⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = { 𝑧 ∣ ∀ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 }
4	2	nfcri	⊢ Ⅎ 𝑦 𝑧 ∈ 𝐵
5	1 4	nfralw	⊢ Ⅎ 𝑦 ∀ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵
6	5	nfab	⊢ Ⅎ 𝑦 { 𝑧 ∣ ∀ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 }
7	3 6	nfcxfr	⊢ Ⅎ 𝑦 ∩ 𝑥 ∈ 𝐴 𝐵