Theorem nfun

Description: Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003) (Revised by Mario Carneiro, 14-Oct-2016)

	Ref	Expression
Hypotheses	nfun.1	⊢ Ⅎ 𝑥 𝐴
	nfun.2	⊢ Ⅎ 𝑥 𝐵
Assertion	nfun	⊢ Ⅎ 𝑥 ( 𝐴 ∪ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		nfun.1	⊢ Ⅎ 𝑥 𝐴
2		nfun.2	⊢ Ⅎ 𝑥 𝐵
3		df-un	⊢ ( 𝐴 ∪ 𝐵 ) = { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵 ) }
4	1	nfcri	⊢ Ⅎ 𝑥 𝑦 ∈ 𝐴
5	2	nfcri	⊢ Ⅎ 𝑥 𝑦 ∈ 𝐵
6	4 5	nfor	⊢ Ⅎ 𝑥 ( 𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵 )
7	6	nfab	⊢ Ⅎ 𝑥 { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵 ) }
8	3 7	nfcxfr	⊢ Ⅎ 𝑥 ( 𝐴 ∪ 𝐵 )