MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ertr2d Structured version   Visualization version   GIF version

Theorem ertr2d 7913
Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
ersymb.1 (𝜑𝑅 Er 𝑋)
ertrd.5 (𝜑𝐴𝑅𝐵)
ertrd.6 (𝜑𝐵𝑅𝐶)
Assertion
Ref Expression
ertr2d (𝜑𝐶𝑅𝐴)

Proof of Theorem ertr2d
StepHypRef Expression
1 ersymb.1 . 2 (𝜑𝑅 Er 𝑋)
2 ertrd.5 . . 3 (𝜑𝐴𝑅𝐵)
3 ertrd.6 . . 3 (𝜑𝐵𝑅𝐶)
41, 2, 3ertrd 7912 . 2 (𝜑𝐴𝑅𝐶)
51, 4ersym 7908 1 (𝜑𝐶𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   class class class wbr 4786   Er wer 7893
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  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-er 7896
This theorem is referenced by:  pi1xfrcnvlem  23075
  Copyright terms: Public domain W3C validator