Theorem nfiund

Description: Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019) Add disjoint variable condition to avoid ax-13 . See nfiundg for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024)

	Ref	Expression
Hypotheses	nfiund.1	⊢ Ⅎ 𝑥 𝜑
	nfiund.2	⊢ ( 𝜑 → Ⅎ 𝑦 𝐴 )
	nfiund.3	⊢ ( 𝜑 → Ⅎ 𝑦 𝐵 )
Assertion	nfiund	⊢ ( 𝜑 → Ⅎ 𝑦 ∪ 𝑥 ∈ 𝐴 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		nfiund.1	⊢ Ⅎ 𝑥 𝜑
2		nfiund.2	⊢ ( 𝜑 → Ⅎ 𝑦 𝐴 )
3		nfiund.3	⊢ ( 𝜑 → Ⅎ 𝑦 𝐵 )
4		df-iun	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 }
5		nfv	⊢ Ⅎ 𝑧 𝜑
6	3	nfcrd	⊢ ( 𝜑 → Ⅎ 𝑦 𝑧 ∈ 𝐵 )
7	1 2 6	nfrexd	⊢ ( 𝜑 → Ⅎ 𝑦 ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 )
8	5 7	nfabdw	⊢ ( 𝜑 → Ⅎ 𝑦 { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 } )
9	4 8	nfcxfrd	⊢ ( 𝜑 → Ⅎ 𝑦 ∪ 𝑥 ∈ 𝐴 𝐵 )