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Theorem equtr2 1941
 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.)
Assertion
Ref Expression
equtr2 ((𝑥 = 𝑧𝑦 = 𝑧) → 𝑥 = 𝑦)

Proof of Theorem equtr2
StepHypRef Expression
1 equeucl 1938 . 2 (𝑥 = 𝑧 → (𝑦 = 𝑧𝑥 = 𝑦))
21imp 444 1 ((𝑥 = 𝑧𝑦 = 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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