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Theorem equtr2 2024
Description: Equality is a left-Euclidean binary relation. Uncurried (imported) form of equeucl 2021. (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 2021 . 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 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:  nfeqf  2384  mo3  2562  mo4  2564  madurid  22666  dchrisumlema  27547  funpartfun  35925  wl-mo3t  37557
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