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| Mirrors > Home > MPE Home > Th. List > equeucl | Structured version Visualization version GIF version | ||
| Description: Equality is a left-Euclidean binary relation. (Right-Euclideanness is stated in ax-7 2006.) Curried (exported) form of equtr2 2025. (Contributed by BJ, 11-Apr-2021.) | 
| Ref | Expression | 
|---|---|
| equeucl | ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | equeuclr 2021 | . 2 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑧 → 𝑥 = 𝑦)) | |
| 2 | 1 | com12 32 | 1 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 | 
| 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: equtr2 2025 sbequ1 2247 ax13lem1 2378 ax13lem2 2380 bj-ax6elem2 36669 wl-ax13lem1 37496 | 
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