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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 2005.) Curried (exported) form of equtr2 2024. (Contributed by BJ, 11-Apr-2021.) |
Ref | Expression |
---|---|
equeucl | ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equeuclr 2020 | . 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 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 |
This theorem is referenced by: equtr2 2024 sbequ1 2246 ax13lem1 2377 ax13lem2 2379 bj-ax6elem2 36650 wl-ax13lem1 37477 |
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