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) Avoid ax-13 . (Revised by GG, 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	2	nfcri	⊢ Ⅎ 𝑥 𝑦 ∈ 𝐴
5	4 1	nfan	⊢ Ⅎ 𝑥 ( 𝑦 ∈ 𝐴 ∧ 𝜑 )
6	5	nfab	⊢ Ⅎ 𝑥 { 𝑦 ∣ ( 𝑦 ∈ 𝐴 ∧ 𝜑 ) }
7	3 6	nfcxfr	⊢ Ⅎ 𝑥 { 𝑦 ∈ 𝐴 ∣ 𝜑 }