Theorem nfriota

Description: A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011)

	Ref	Expression
Hypotheses	nfriota.1	⊢ Ⅎ 𝑥 𝜑
	nfriota.2	⊢ Ⅎ 𝑥 𝐴
Assertion	nfriota	⊢ Ⅎ 𝑥 ( ℩ 𝑦 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		nfriota.1	⊢ Ⅎ 𝑥 𝜑
2		nfriota.2	⊢ Ⅎ 𝑥 𝐴
3		nftru	⊢ Ⅎ 𝑦 ⊤
4	1	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝜑 )
5	2	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝐴 )
6	3 4 5	nfriotadw	⊢ ( ⊤ → Ⅎ 𝑥 ( ℩ 𝑦 ∈ 𝐴 𝜑 ) )
7	6	mptru	⊢ Ⅎ 𝑥 ( ℩ 𝑦 ∈ 𝐴 𝜑 )