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| Description: Variation on nfald 2328 which adds the hypothesis that 𝑥 and 𝑦 are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.) Usage of this theorem is discouraged because it depends on ax-13 2377. Use nfald 2328 instead. (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| nfald2.1 | ⊢ Ⅎ𝑦𝜑 | 
| nfald2.2 | ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | 
| Ref | Expression | 
|---|---|
| nfald2 | ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfald2.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
| 2 | nfnae 2439 | . . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦 | |
| 3 | 1, 2 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) | 
| 4 | nfald2.2 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | |
| 5 | 3, 4 | nfald 2328 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥∀𝑦𝜓) | 
| 6 | 5 | ex 412 | . 2 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑦𝜓)) | 
| 7 | nfa1 2151 | . . 3 ⊢ Ⅎ𝑦∀𝑦𝜓 | |
| 8 | biidd 262 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 ↔ ∀𝑦𝜓)) | |
| 9 | 8 | drnf1 2448 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥∀𝑦𝜓 ↔ Ⅎ𝑦∀𝑦𝜓)) | 
| 10 | 7, 9 | mpbiri 258 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑦𝜓) | 
| 11 | 6, 10 | pm2.61d2 181 | 1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1538 Ⅎwnf 1783 | 
| 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-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 | 
| This theorem is referenced by: nfexd2 2451 dvelimf 2453 nfmod2 2558 nfrald 3372 nfiotad 6519 nfixp 8957 | 
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