Theorem nfralw

Description: Bound-variable hypothesis builder for restricted quantification. Version of nfral with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 1-Sep-1999) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypotheses	nfralw.1	⊢ Ⅎ 𝑥 𝐴
	nfralw.2	⊢ Ⅎ 𝑥 𝜑
Assertion	nfralw	⊢ Ⅎ 𝑥 ∀ 𝑦 ∈ 𝐴 𝜑

Proof

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