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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 38871. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 2154. (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 2149 | . . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑥 | |
2 | nfna1 2149 | . . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦 | |
3 | 1, 2 | nfan 1896 | . 2 ⊢ Ⅎ𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) |
4 | equvinva 2026 | . . 3 ⊢ (𝑥 = 𝑦 → ∃𝑤(𝑥 = 𝑤 ∧ 𝑦 = 𝑤)) | |
5 | dveeq1 2382 | . . . . . . . 8 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑤 → ∀𝑧 𝑥 = 𝑤)) | |
6 | 5 | imp 406 | . . . . . . 7 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ 𝑥 = 𝑤) → ∀𝑧 𝑥 = 𝑤) |
7 | dveeq1 2382 | . . . . . . . 8 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → (𝑦 = 𝑤 → ∀𝑧 𝑦 = 𝑤)) | |
8 | 7 | imp 406 | . . . . . . 7 ⊢ ((¬ ∀𝑧 𝑧 = 𝑦 ∧ 𝑦 = 𝑤) → ∀𝑧 𝑦 = 𝑤) |
9 | equtr2 2023 | . . . . . . . 8 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑤) → 𝑥 = 𝑦) | |
10 | 9 | alanimi 1812 | . . . . . . 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 1930 | . . 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 2282 | 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 1534 ∃wex 1775 Ⅎwnf 1779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-10 2138 ax-12 2174 ax-13 2374 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-ex 1776 df-nf 1780 |
This theorem is referenced by: axc9 2384 dvelimf 2450 equvel 2458 2ax6elem 2472 wl-exeq 37514 wl-nfeqfb 37516 wl-equsb4 37537 wl-2sb6d 37538 wl-sbalnae 37542 |
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