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Mirrors > Home > ILE Home > Th. List > euequ1 | GIF version |
Description: Equality has existential uniqueness. (Contributed by Stefan Allan, 4-Dec-2008.) |
Ref | Expression |
---|---|
euequ1 | ⊢ ∃!𝑥 𝑥 = 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | a9e 1689 | . 2 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | equtr2 1704 | . . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑧) | |
3 | 2 | gen2 1443 | . 2 ⊢ ∀𝑥∀𝑧((𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑧) |
4 | equequ1 1705 | . . 3 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑧 = 𝑦)) | |
5 | 4 | eu4 2081 | . 2 ⊢ (∃!𝑥 𝑥 = 𝑦 ↔ (∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥∀𝑧((𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑧))) |
6 | 1, 3, 5 | mpbir2an 937 | 1 ⊢ ∃!𝑥 𝑥 = 𝑦 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1346 ∃wex 1485 ∃!weu 2019 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 |
This theorem is referenced by: copsexg 4227 oprabid 5883 |
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