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Mirrors > Home > ILE Home > Th. List > mo2n | GIF version |
Description: There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.) |
Ref | Expression |
---|---|
mon.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
mo2n | ⊢ (¬ ∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mon.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | sb8e 1809 | . 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
3 | alnex 1456 | . . 3 ⊢ (∀𝑦 ¬ [𝑦 / 𝑥]𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑) | |
4 | nfs1v 1888 | . . . . . 6 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
5 | 4 | nfn 1617 | . . . . 5 ⊢ Ⅎ𝑥 ¬ [𝑦 / 𝑥]𝜑 |
6 | 1 | nfn 1617 | . . . . 5 ⊢ Ⅎ𝑦 ¬ 𝜑 |
7 | sbequ1 1722 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑)) | |
8 | 7 | equcoms 1665 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜑 → [𝑦 / 𝑥]𝜑)) |
9 | 8 | con3d 603 | . . . . 5 ⊢ (𝑦 = 𝑥 → (¬ [𝑦 / 𝑥]𝜑 → ¬ 𝜑)) |
10 | 5, 6, 9 | cbv3 1701 | . . . 4 ⊢ (∀𝑦 ¬ [𝑦 / 𝑥]𝜑 → ∀𝑥 ¬ 𝜑) |
11 | pm2.21 589 | . . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝑥 = 𝑦)) | |
12 | 11 | alimi 1412 | . . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑦)) |
13 | 19.8a 1550 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | |
14 | 10, 12, 13 | 3syl 17 | . . 3 ⊢ (∀𝑦 ¬ [𝑦 / 𝑥]𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
15 | 3, 14 | sylbir 134 | . 2 ⊢ (¬ ∃𝑦[𝑦 / 𝑥]𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
16 | 2, 15 | sylnbi 650 | 1 ⊢ (¬ ∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1310 Ⅎwnf 1417 ∃wex 1449 [wsb 1716 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-11 1465 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 |
This theorem depends on definitions: df-bi 116 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 |
This theorem is referenced by: modc 2016 |
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