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Theorem equtr2 1951
 Description: Equality is a left-Euclidean binary relation. Uncurried (imported) form of equeucl 1948. (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 1948 . 2 (𝑥 = 𝑧 → (𝑦 = 𝑧𝑥 = 𝑦))
21imp 445 1 ((𝑥 = 𝑧𝑦 = 𝑧) → 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702 This theorem is referenced by:  nfeqf  2300  mo3  2506  madurid  20378  dchrisumlema  25090  funpartfun  31719  bj-ssbequ1  32313  bj-mo3OLD  32504  wl-mo3t  33017
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