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

Theorem tprot 4259
Description: Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
tprot {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}

Proof of Theorem tprot
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 3orrot 1042 . . 3 ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) ↔ (𝑥 = 𝐵𝑥 = 𝐶𝑥 = 𝐴))
21abbii 2736 . 2 {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)} = {𝑥 ∣ (𝑥 = 𝐵𝑥 = 𝐶𝑥 = 𝐴)}
3 dftp2 4207 . 2 {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}
4 dftp2 4207 . 2 {𝐵, 𝐶, 𝐴} = {𝑥 ∣ (𝑥 = 𝐵𝑥 = 𝐶𝑥 = 𝐴)}
52, 3, 43eqtr4i 2653 1 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
Colors of variables: wff setvar class
Syntax hints:  w3o 1035   = wceq 1480  {cab 2607  {ctp 4157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-un 3564  df-sn 4154  df-pr 4156  df-tp 4158
This theorem is referenced by:  tpcomb  4261  tpass  4262  tpidm13  4266  tpidm23  4267  tpnzd  4289  tpprceq3  4309  fvtp2  6421  fvtp3  6422  fvtp2g  6424  fvtp3g  6425  f13dfv  6490  en3lplem2  8463  estrres  16707  nb3grprlem2  26183  nb3grpr  26184  nb3grpr2  26185  nb3gr2nb  26186  cplgr3v  26231  frgr3v  27016  1to3vfriswmgr  27021  dvh4dimN  36243  en3lplem2VD  38589
  Copyright terms: Public domain W3C validator