Theorem nfraldw 3148

Description: Deduction version of nfralw 3151. Version of nfrald 3150 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by NM, 15-Feb-2013.) (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 3070	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
2		nfraldw.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfraldw.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4	3	nfcrd 2897	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
5		nfraldw.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
6	4, 5	nfimd 1900	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓))
7	2, 6	nfald 2325	. 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
8	1, 7	nfxfrd 1859	1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4  ∀wal 1539  Ⅎwnf 1789   ∈ wcel 2109  Ⅎwnfc 2888  ∀wral 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-10 2140  ax-11 2157  ax-12 2174

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1786  df-nf 1790  df-clel 2817  df-nfc 2890  df-ral 3070

This theorem is referenced by:  nfralw  3151  nfrexd  3237  nfttrcld  9429