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Theorem equtr2 2025
Description: Equality is a left-Euclidean binary relation. Uncurried (imported) form of equeucl 2022. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by BJ, 11-Apr-2021.)
Assertion
Ref Expression
equtr2 ((𝑥 = 𝑧𝑦 = 𝑧) → 𝑥 = 𝑦)

Proof of Theorem equtr2
StepHypRef Expression
1 equeucl 2022 . 2 (𝑥 = 𝑧 → (𝑦 = 𝑧𝑥 = 𝑦))
21imp 406 1 ((𝑥 = 𝑧𝑦 = 𝑧) → 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395
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:  nfeqf  2385  mo3  2563  mo4  2565  madurid  22651  dchrisumlema  27533  funpartfun  35945  wl-mo3t  37578
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