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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 1868 | . 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
3 | alnex 1510 | . . 3 ⊢ (∀𝑦 ¬ [𝑦 / 𝑥]𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑) | |
4 | nfs1v 1955 | . . . . . 6 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
5 | 4 | nfn 1669 | . . . . 5 ⊢ Ⅎ𝑥 ¬ [𝑦 / 𝑥]𝜑 |
6 | 1 | nfn 1669 | . . . . 5 ⊢ Ⅎ𝑦 ¬ 𝜑 |
7 | sbequ1 1779 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑)) | |
8 | 7 | equcoms 1719 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜑 → [𝑦 / 𝑥]𝜑)) |
9 | 8 | con3d 632 | . . . . 5 ⊢ (𝑦 = 𝑥 → (¬ [𝑦 / 𝑥]𝜑 → ¬ 𝜑)) |
10 | 5, 6, 9 | cbv3 1753 | . . . 4 ⊢ (∀𝑦 ¬ [𝑦 / 𝑥]𝜑 → ∀𝑥 ¬ 𝜑) |
11 | pm2.21 618 | . . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝑥 = 𝑦)) | |
12 | 11 | alimi 1466 | . . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑦)) |
13 | 19.8a 1601 | . . . 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 135 | . 2 ⊢ (¬ ∃𝑦[𝑦 / 𝑥]𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
16 | 2, 15 | sylnbi 679 | 1 ⊢ (¬ ∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1362 Ⅎwnf 1471 ∃wex 1503 [wsb 1773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 |
This theorem is referenced by: modc 2085 |
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