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Metamath Proof Explorer

Theorem nfiung

Description: Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 . See nfiun for a version with more disjoint variable conditions, but not requiring ax-13 . (Contributed by Mario Carneiro, 25-Jan-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfiung.1	⊢ Ⅎ 𝑦 𝐴
	nfiung.2	⊢ Ⅎ 𝑦 𝐵
Assertion	nfiung	⊢ Ⅎ 𝑦 ∪ 𝑥 ∈ 𝐴 𝐵

Proof

Step	Hyp	Ref	Expression
1		nfiung.1	⊢ Ⅎ 𝑦 𝐴
2		nfiung.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	nfabg	⊢ Ⅎ 𝑦 { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 }
7	3 6	nfcxfr	⊢ Ⅎ 𝑦 ∪ 𝑥 ∈ 𝐴 𝐵