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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 2031. (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 2031 | . 2 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦)) | |
2 | 1 | imp 409 | 1 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 |
This theorem is referenced by: nfeqf 2399 mo3 2648 mo4 2650 madurid 21253 dchrisumlema 26064 funpartfun 33404 wl-mo3t 34827 |
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