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Metamath Proof Explorer

Theorem nfiu1

Description: Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003) Avoid ax-11 , ax-12 . (Revised by SN, 14-May-2025)

		Ref	Expression
	Assertion	nfiu1	⊢ Ⅎ 𝑥 ∪ 𝑥 ∈ 𝐴 𝐵

Proof

Step	Hyp	Ref	Expression
1		eliun	⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 )
2		nfre1	⊢ Ⅎ 𝑥 ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
3	1 2	nfxfr	⊢ Ⅎ 𝑥 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵
4	3	nfci	⊢ Ⅎ 𝑥 ∪ 𝑥 ∈ 𝐴 𝐵