Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nmo | Structured version Visualization version GIF version |
Description: Negation of "at most one". (Contributed by Thierry Arnoux, 26-Feb-2017.) |
Ref | Expression |
---|---|
nmo.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
nmo | ⊢ (¬ ∃*𝑥𝜑 ↔ ∀𝑦∃𝑥(𝜑 ∧ 𝑥 ≠ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmo.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | mof 2647 | . . 3 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
3 | 2 | notbii 322 | . 2 ⊢ (¬ ∃*𝑥𝜑 ↔ ¬ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
4 | alnex 1782 | . 2 ⊢ (∀𝑦 ¬ ∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ¬ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | |
5 | exnal 1827 | . . . 4 ⊢ (∃𝑥 ¬ (𝜑 → 𝑥 = 𝑦) ↔ ¬ ∀𝑥(𝜑 → 𝑥 = 𝑦)) | |
6 | pm4.61 407 | . . . . . 6 ⊢ (¬ (𝜑 → 𝑥 = 𝑦) ↔ (𝜑 ∧ ¬ 𝑥 = 𝑦)) | |
7 | biid 263 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦) | |
8 | 7 | necon3bbii 3065 | . . . . . . 7 ⊢ (¬ 𝑥 = 𝑦 ↔ 𝑥 ≠ 𝑦) |
9 | 8 | anbi2i 624 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑥 = 𝑦) ↔ (𝜑 ∧ 𝑥 ≠ 𝑦)) |
10 | 6, 9 | bitri 277 | . . . . 5 ⊢ (¬ (𝜑 → 𝑥 = 𝑦) ↔ (𝜑 ∧ 𝑥 ≠ 𝑦)) |
11 | 10 | exbii 1848 | . . . 4 ⊢ (∃𝑥 ¬ (𝜑 → 𝑥 = 𝑦) ↔ ∃𝑥(𝜑 ∧ 𝑥 ≠ 𝑦)) |
12 | 5, 11 | bitr3i 279 | . . 3 ⊢ (¬ ∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∃𝑥(𝜑 ∧ 𝑥 ≠ 𝑦)) |
13 | 12 | albii 1820 | . 2 ⊢ (∀𝑦 ¬ ∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑦∃𝑥(𝜑 ∧ 𝑥 ≠ 𝑦)) |
14 | 3, 4, 13 | 3bitr2i 301 | 1 ⊢ (¬ ∃*𝑥𝜑 ↔ ∀𝑦∃𝑥(𝜑 ∧ 𝑥 ≠ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 ∃wex 1780 Ⅎwnf 1784 ∃*wmo 2620 ≠ wne 3018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-11 2161 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-mo 2622 df-ne 3019 |
This theorem is referenced by: (None) |
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