Theorem nfraldw 3307

Description: Deduction version of nfralw 3309. Version of nfrald 3369 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 15-Feb-2013.) Avoid ax-9 2117, ax-ext 2704. (Revised by Gino Giotto, 24-Sep-2024.)

Hypotheses

Ref	Expression
nfraldw.1	⊢ Ⅎ𝑦𝜑
nfraldw.2	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfraldw.3	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfraldw	⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfraldw

Step	Hyp	Ref	Expression
1		df-ral 3063	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
2		nfraldw.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfraldw.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4	3	nfcrd 2893	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
5		nfraldw.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
6	4, 5	nfimd 1898	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓))
7	2, 6	nfald 2322	. 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
8	1, 7	nfxfrd 1857	1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4  ∀wal 1540  Ⅎwnf 1786   ∈ wcel 2107  Ⅎwnfc 2884  ∀wral 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-10 2138  ax-11 2155  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-clel 2811  df-nfc 2886  df-ral 3063

This theorem is referenced by:  nfrexdw  3308  nfralwOLD  3310  nfttrcld  9705