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Theorem equeucl 2120
Description: Equality is a left-Euclidean binary relation. (Right-Euclideanness is stated in ax-7 2104.) Curried (exported) form of equtr2 2123. (Contributed by BJ, 11-Apr-2021.)
Assertion
Ref Expression
equeucl (𝑥 = 𝑧 → (𝑦 = 𝑧𝑥 = 𝑦))

Proof of Theorem equeucl
StepHypRef Expression
1 equeuclr 2119 . 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 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1860
This theorem is referenced by:  equtr2  2123  ax13lem1  2422  ax13lem2  2463  bj-ax6elem2  32965  wl-ax13lem1  33603
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