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

Theorem equeuclr 1947
Description: Commuted version of equeucl 1948 (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 1946 . 2 (𝑧 = 𝑥 → (𝑦 = 𝑧𝑦 = 𝑥))
21equcoms 1944 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 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:  equeucl  1948  equequ2  1950  ax13b  1961  aevlem0  1977  equvini  2345  sbequi  2374  wl-ax8clv2  33048
  Copyright terms: Public domain W3C validator