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

Theorem equtrr 2019
Description: A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)
Assertion
Ref Expression
equtrr (𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))

Proof of Theorem equtrr
StepHypRef Expression
1 equtr 2018 . 2 (𝑧 = 𝑥 → (𝑥 = 𝑦𝑧 = 𝑦))
21com12 32 1 (𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 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 1908  ax-6 1965  ax-7 2005
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777
This theorem is referenced by:  equeuclr  2020  equequ2  2023  ax12v2  2177  2ax6elem  2473  axprlem3OLD  5434  wl-spae  37502  ax12eq  38923  sn-axprlem3  42235  ax6e2eq  44555  ax6e2eqVD  44905
  Copyright terms: Public domain W3C validator