Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2036. (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 2036 | . 2 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦)) | |
2 | 1 | imp 410 | 1 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1787 |
This theorem is referenced by: nfeqf 2382 mo3 2565 mo4 2567 madurid 21407 dchrisumlema 26236 funpartfun 33900 wl-mo3t 35386 |
Copyright terms: Public domain | W3C validator |