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| Mirrors > Home > MPE Home > Th. List > nfmod2 | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for the at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 2377. See nfmodv 2559 for a version replacing the distinctor with a disjoint variable condition, not requiring ax-13 2377. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid df-eu 2569. (Revised by BJ, 14-Oct-2022.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nfmod2.1 | ⊢ Ⅎ𝑦𝜑 |
| nfmod2.2 | ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
| Ref | Expression |
|---|---|
| nfmod2 | ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-mo 2540 | . 2 ⊢ (∃*𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) | |
| 2 | nfv 1914 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
| 3 | nfmod2.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 4 | nfmod2.2 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | |
| 5 | nfeqf1 2384 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧) | |
| 6 | 5 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 = 𝑧) |
| 7 | 4, 6 | nfimd 1894 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝜓 → 𝑦 = 𝑧)) |
| 8 | 3, 7 | nfald2 2450 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 → 𝑦 = 𝑧)) |
| 9 | 2, 8 | nfexd 2329 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) |
| 10 | 1, 9 | nfxfrd 1854 | 1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1538 ∃wex 1779 Ⅎwnf 1783 ∃*wmo 2538 |
| 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 df-mo 2540 |
| This theorem is referenced by: nfmod 2561 nfeud2 2590 nfrmod 3432 nfrmo 3434 nfdisj 5123 |
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