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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. Imported (uncurried) form of equeucl 1938. (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 1938 | . 2 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦)) | |
2 | 1 | imp 444 | 1 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: nfeqf 2289 mo3 2495 fundmge2nop0 13129 madurid 20269 dchrisumlema 24977 funpartfun 31220 bj-ssbequ1 31833 bj-mo3OLD 32022 wl-mo3t 32537 |
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