Theorem nfrabw

Description: A variable not free in a wff remains so in a restricted class abstraction. Version of nfrab with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 13-Oct-2003) (Revised by Gino Giotto, 10-Jan-2024) (Proof shortened by Wolf Lammen, 23-Nov-2024)

	Ref	Expression
Hypotheses	nfrabw.1	⊢ Ⅎ 𝑥 𝜑
	nfrabw.2	⊢ Ⅎ 𝑥 𝐴
Assertion	nfrabw	⊢ Ⅎ 𝑥 { 𝑦 ∈ 𝐴 ∣ 𝜑 }

Proof

Step	Hyp	Ref	Expression
1		nfrabw.1	⊢ Ⅎ 𝑥 𝜑
2		nfrabw.2	⊢ Ⅎ 𝑥 𝐴
3		df-rab	⊢ { 𝑦 ∈ 𝐴 ∣ 𝜑 } = { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) }
4		nftru	⊢ Ⅎ 𝑦 ⊤
5	2	nfcri	⊢ Ⅎ 𝑥 𝑦 ∈ 𝐴
6	5 1	nfan	⊢ Ⅎ 𝑥 ( 𝑦 ∈ 𝐴 ∧ 𝜑 )
7	6	a1i	⊢ ( ⊤ → Ⅎ 𝑥 ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) )
8	4 7	nfabdw	⊢ ( ⊤ → Ⅎ 𝑥 { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) } )
9	8	mptru	⊢ Ⅎ 𝑥 { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) }
10	3 9	nfcxfr	⊢ Ⅎ 𝑥 { 𝑦 ∈ 𝐴 ∣ 𝜑 }