Theorem nfiun

Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014) Add disjoint variable condition to avoid ax-13 . See nfiung 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	nfiun	⊢ Ⅎ 𝑦 ∪ 𝑥 ∈ 𝐴 𝐵

Proof

Step	Hyp	Ref	Expression
1		nfiun.1	⊢ Ⅎ 𝑦 𝐴
2		nfiun.2	⊢ Ⅎ 𝑦 𝐵
3		df-iun	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 }
4	2	nfcri	⊢ Ⅎ 𝑦 𝑧 ∈ 𝐵
5	1 4	nfrex	⊢ 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