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Mirrors > Home > MPE Home > Th. List > nfald2 | Structured version Visualization version GIF version |
Description: Variation on nfald 2321 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 2371. Use nfald 2321 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 2433 | . . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦 | |
3 | 1, 2 | nfan 1902 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) |
4 | nfald2.2 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | |
5 | 3, 4 | nfald 2321 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥∀𝑦𝜓) |
6 | 5 | ex 413 | . 2 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑦𝜓)) |
7 | nfa1 2148 | . . 3 ⊢ Ⅎ𝑦∀𝑦𝜓 | |
8 | biidd 261 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 ↔ ∀𝑦𝜓)) | |
9 | 8 | drnf1 2442 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥∀𝑦𝜓 ↔ Ⅎ𝑦∀𝑦𝜓)) |
10 | 7, 9 | mpbiri 257 | . 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 396 ∀wal 1539 Ⅎwnf 1785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-11 2154 ax-12 2171 ax-13 2371 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 |
This theorem is referenced by: nfexd2 2445 dvelimf 2447 nfmod2 2552 nfrald 3368 nfiotad 6497 nfixp 8907 |
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