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| Description: Deduction version of nfral 3374. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker nfraldw 3309 when possible. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| nfrald.1 | ⊢ Ⅎ𝑦𝜑 | 
| nfrald.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) | 
| nfrald.3 | ⊢ (𝜑 → Ⅎ𝑥𝜓) | 
| Ref | Expression | 
|---|---|
| nfrald | ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-ral 3062 | . 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) | |
| 2 | nfrald.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
| 3 | nfcvf 2932 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
| 4 | 3 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦) | 
| 5 | nfrald.2 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
| 6 | 5 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴) | 
| 7 | 4, 6 | nfeld 2917 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴) | 
| 8 | nfrald.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
| 9 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | 
| 10 | 7, 9 | nfimd 1894 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓)) | 
| 11 | 2, 10 | nfald2 2450 | . 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) | 
| 12 | 1, 11 | nfxfrd 1854 | 1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1538 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 ∀wral 3061 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 | 
| This theorem is referenced by: nfrexd 3373 nfral 3374 | 
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