Theorem nfraldw 3187

Description: Deduction version of nfralw 3189. Version of nfrald 3188 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-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 3111	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
2		nfraldw.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfcvd 2956	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦)
4		nfraldw.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5	3, 4	nfeld 2966	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
6		nfraldw.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
7	5, 6	nfimd 1895	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓))
8	2, 7	nfald 2336	. 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
9	1, 8	nfxfrd 1855	1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4  ∀wal 1536  Ⅎwnf 1785   ∈ wcel 2111  Ⅎwnfc 2936  ∀wral 3106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111

This theorem is referenced by:  nfralw  3189  nfrexd  3266