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

Theorem tpcomb 4318
Description: Swap 2nd and 3rd members of an unordered triple. (Contributed by NM, 22-May-2015.)
Assertion
Ref Expression
tpcomb {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐵}

Proof of Theorem tpcomb
StepHypRef Expression
1 tpcoma 4317 . 2 {𝐵, 𝐶, 𝐴} = {𝐶, 𝐵, 𝐴}
2 tprot 4316 . 2 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
3 tprot 4316 . 2 {𝐴, 𝐶, 𝐵} = {𝐶, 𝐵, 𝐴}
41, 2, 33eqtr4i 2683 1 {𝐴, 𝐵, 𝐶} = {𝐴, 𝐶, 𝐵}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  {ctp 4214
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  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-sn 4211  df-pr 4213  df-tp 4215
This theorem is referenced by:  f13dfv  6570  frgr3v  27255  signswch  30766  signstfvcl  30778  dvh4dimN  37053
  Copyright terms: Public domain W3C validator