MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  equeuclr Structured version   Visualization version   GIF version

Theorem equeuclr 2108
Description: Commuted version of equeucl 2109 (equality is left-Euclidean). (Contributed by BJ, 12-Apr-2021.)
Assertion
Ref Expression
equeuclr (𝑥 = 𝑧 → (𝑦 = 𝑧𝑦 = 𝑥))

Proof of Theorem equeuclr
StepHypRef Expression
1 equtrr 2107 . 2 (𝑧 = 𝑥 → (𝑦 = 𝑧𝑦 = 𝑥))
21equcoms 2105 1 (𝑥 = 𝑧 → (𝑦 = 𝑧𝑦 = 𝑥))
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 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853
This theorem is referenced by:  equeucl  2109  equequ2  2111  ax13b  2120  aevlem0  2137  equvini  2492  sbequi  2522  wl-ax8clv2  33714
  Copyright terms: Public domain W3C validator