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| Mirrors > Home > MPE Home > Th. List > nfeqf | Structured version Visualization version GIF version | ||
| Description: A variable is effectively not free in an equality if it is not either of the involved variables. Ⅎ version of ax-c9 38891. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 2157. (Revised by Wolf Lammen, 6-Sep-2018.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nfeqf | ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfna1 2152 | . . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑥 | |
| 2 | nfna1 2152 | . . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦 | |
| 3 | 1, 2 | nfan 1899 | . 2 ⊢ Ⅎ𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) |
| 4 | equvinva 2029 | . . 3 ⊢ (𝑥 = 𝑦 → ∃𝑤(𝑥 = 𝑤 ∧ 𝑦 = 𝑤)) | |
| 5 | dveeq1 2385 | . . . . . . . 8 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑤 → ∀𝑧 𝑥 = 𝑤)) | |
| 6 | 5 | imp 406 | . . . . . . 7 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ 𝑥 = 𝑤) → ∀𝑧 𝑥 = 𝑤) |
| 7 | dveeq1 2385 | . . . . . . . 8 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → (𝑦 = 𝑤 → ∀𝑧 𝑦 = 𝑤)) | |
| 8 | 7 | imp 406 | . . . . . . 7 ⊢ ((¬ ∀𝑧 𝑧 = 𝑦 ∧ 𝑦 = 𝑤) → ∀𝑧 𝑦 = 𝑤) |
| 9 | equtr2 2026 | . . . . . . . 8 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑤) → 𝑥 = 𝑦) | |
| 10 | 9 | alanimi 1816 | . . . . . . 7 ⊢ ((∀𝑧 𝑥 = 𝑤 ∧ ∀𝑧 𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦) |
| 11 | 6, 8, 10 | syl2an 596 | . . . . . 6 ⊢ (((¬ ∀𝑧 𝑧 = 𝑥 ∧ 𝑥 = 𝑤) ∧ (¬ ∀𝑧 𝑧 = 𝑦 ∧ 𝑦 = 𝑤)) → ∀𝑧 𝑥 = 𝑦) |
| 12 | 11 | an4s 660 | . . . . 5 ⊢ (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) ∧ (𝑥 = 𝑤 ∧ 𝑦 = 𝑤)) → ∀𝑧 𝑥 = 𝑦) |
| 13 | 12 | ex 412 | . . . 4 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ((𝑥 = 𝑤 ∧ 𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦)) |
| 14 | 13 | exlimdv 1933 | . . 3 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∃𝑤(𝑥 = 𝑤 ∧ 𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦)) |
| 15 | 4, 14 | syl5 34 | . 2 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) |
| 16 | 3, 15 | nf5d 2284 | 1 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1538 ∃wex 1779 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-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: axc9 2387 dvelimf 2453 equvel 2461 2ax6elem 2475 wl-exeq 37535 wl-nfeqfb 37537 wl-equsb4 37558 wl-2sb6d 37559 wl-sbalnae 37563 |
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