Theorem nfraldw

Description: Deduction version of nfralw . Version of nfrald with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 15-Feb-2013) (Revised by Gino Giotto, 24-Sep-2024)

	Ref	Expression
Hypotheses	nfraldw.1	⊢ Ⅎ 𝑦 𝜑
	nfraldw.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
	nfraldw.3	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
Assertion	nfraldw	⊢ ( 𝜑 → Ⅎ 𝑥 ∀ 𝑦 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		nfraldw.1	⊢ Ⅎ 𝑦 𝜑
2		nfraldw.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
3		nfraldw.3	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
4		df-ral	⊢ ( ∀ 𝑦 ∈ 𝐴 𝜓 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝜓 ) )
5	2	nfcrd	⊢ ( 𝜑 → Ⅎ 𝑥 𝑦 ∈ 𝐴 )
6	5 3	nfimd	⊢ ( 𝜑 → Ⅎ 𝑥 ( 𝑦 ∈ 𝐴 → 𝜓 ) )
7	1 6	nfald	⊢ ( 𝜑 → Ⅎ 𝑥 ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝜓 ) )
8	4 7	nfxfrd	⊢ ( 𝜑 → Ⅎ 𝑥 ∀ 𝑦 ∈ 𝐴 𝜓 )