Theorem nfraldw 3296

Description: Deduction version of nfralw 3298. Version of nfrald 3355 with a disjoint variable condition, which does not require ax-13 2365. (Contributed by NM, 15-Feb-2013.) Avoid ax-9 2108, ax-ext 2696. (Revised by GG, 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 3051	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
2		nfraldw.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfraldw.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4	3	nfcrd 2884	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
5		nfraldw.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
6	4, 5	nfimd 1889	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓))
7	2, 6	nfald 2316	. 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
8	1, 7	nfxfrd 1848	1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4  ∀wal 1531  Ⅎwnf 1777   ∈ wcel 2098  Ⅎwnfc 2875  ∀wral 3050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-10 2129  ax-11 2146  ax-12 2166

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1774  df-nf 1778  df-clel 2802  df-nfc 2877  df-ral 3051

This theorem is referenced by:  nfrexdw  3297  nfralwOLD  3299  nfttrcld  9740