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Mirrors > Home > MPE Home > Th. List > nfald2 | Structured version Visualization version GIF version |
Description: Variation on nfald 2343 which adds the hypothesis that 𝑥 and 𝑦 are distinct in the inner subproof. Usage of this theorem is discouraged because it depends on ax-13 2386. Check out nfald 2343 for a version requiring fewer axioms. (Contributed by Mario Carneiro, 8-Oct-2016.) (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 2452 | . . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦 | |
3 | 1, 2 | nfan 1896 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) |
4 | nfald2.2 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | |
5 | 3, 4 | nfald 2343 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥∀𝑦𝜓) |
6 | 5 | ex 415 | . 2 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑦𝜓)) |
7 | nfa1 2151 | . . 3 ⊢ Ⅎ𝑦∀𝑦𝜓 | |
8 | biidd 264 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 ↔ ∀𝑦𝜓)) | |
9 | 8 | drnf1 2461 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥∀𝑦𝜓 ↔ Ⅎ𝑦∀𝑦𝜓)) |
10 | 7, 9 | mpbiri 260 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑦𝜓) |
11 | 6, 10 | pm2.61d2 183 | 1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∀wal 1531 Ⅎwnf 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-10 2141 ax-11 2157 ax-12 2173 ax-13 2386 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 |
This theorem is referenced by: nfexd2 2464 dvelimf 2466 nfmod2 2638 nfrald 3224 nfiotad 6313 nfixp 8475 |
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