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

Theorem tpcomb 4124
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 4123 . 2  |-  { B ,  C ,  A }  =  { C ,  B ,  A }
2 tprot 4122 . 2  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
3 tprot 4122 . 2  |-  { A ,  C ,  B }  =  { C ,  B ,  A }
41, 2, 33eqtr4i 2506 1  |-  { A ,  B ,  C }  =  { A ,  C ,  B }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   {ctp 4031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-un 3481  df-sn 4028  df-pr 4030  df-tp 4032
This theorem is referenced by:  f13dfv  6166  cusgra3v  24140  frgra3v  24678  signswch  28158  signstfvcl  28170  dvh4dimN  36244
  Copyright terms: Public domain W3C validator