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Theorem equeucl 2027
Description: Equality is a left-Euclidean binary relation. (Right-Euclideanness is stated in ax-7 2011.) Curried (exported) form of equtr2 2030. (Contributed by BJ, 11-Apr-2021.)
Assertion
Ref Expression
equeucl (𝑥 = 𝑧 → (𝑦 = 𝑧𝑥 = 𝑦))

Proof of Theorem equeucl
StepHypRef Expression
1 equeuclr 2026 . 2 (𝑦 = 𝑧 → (𝑥 = 𝑧𝑥 = 𝑦))
21com12 32 1 (𝑥 = 𝑧 → (𝑦 = 𝑧𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4
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 1907  ax-6 1966  ax-7 2011
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777
This theorem is referenced by:  equtr2  2030  sbequ1  2245  ax13lem1  2388  ax13lem2  2390  bj-ax6elem2  33995  wl-ax13lem1  34740
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