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| Mirrors > Home > MPE Home > Th. List > equtr2 | Structured version Visualization version GIF version | ||
| Description: Equality is a left-Euclidean binary relation. Uncurried (imported) form of equeucl 2022. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by BJ, 11-Apr-2021.) | 
| Ref | Expression | 
|---|---|
| equtr2 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | equeucl 2022 | . 2 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦)) | |
| 2 | 1 | imp 406 | 1 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 | 
| This theorem is referenced by: nfeqf 2385 mo3 2563 mo4 2565 madurid 22651 dchrisumlema 27533 funpartfun 35945 wl-mo3t 37578 | 
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