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| Description: Deduction version of nfralw 3310. Version of nfrald 3371 with a disjoint variable condition, which does not require ax-13 2376. (Contributed by NM, 15-Feb-2013.) Avoid ax-9 2117, ax-ext 2707. (Revised by GG, 24-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| nfraldw.1 | ⊢ Ⅎ𝑦𝜑 | 
| nfraldw.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) | 
| nfraldw.3 | ⊢ (𝜑 → Ⅎ𝑥𝜓) | 
| Ref | Expression | 
|---|---|
| nfraldw | ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-ral 3061 | . 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) | |
| 2 | nfraldw.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
| 3 | nfraldw.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
| 4 | 3 | nfcrd 2898 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) | 
| 5 | nfraldw.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
| 6 | 4, 5 | nfimd 1893 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓)) | 
| 7 | 2, 6 | nfald 2327 | . 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) | 
| 8 | 1, 7 | nfxfrd 1853 | 1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 Ⅎwnf 1782 ∈ wcel 2107 Ⅎwnfc 2889 ∀wral 3060 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-10 2140 ax-11 2156 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1779 df-nf 1783 df-clel 2815 df-nfc 2891 df-ral 3061 | 
| This theorem is referenced by: nfrexdw 3309 nfralwOLD 3311 nfttrcld 9751 | 
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