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

Theorem equtr 1994
Description: A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
Assertion
Ref Expression
equtr (𝑥 = 𝑦 → (𝑦 = 𝑧𝑥 = 𝑧))

Proof of Theorem equtr
StepHypRef Expression
1 ax7 1989 . 2 (𝑦 = 𝑥 → (𝑦 = 𝑧𝑥 = 𝑧))
21equcoms 1993 1 (𝑥 = 𝑦 → (𝑦 = 𝑧𝑥 = 𝑧))
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 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by:  equtrr  1995  equequ1  1998  equviniva  2004  ax6e  2286  equvini  2374  sbequi  2403  axsep  4813  bj-axsep  32918
  Copyright terms: Public domain W3C validator