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Theorem tprot 4228
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 1037 . . 3 ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) ↔ (𝑥 = 𝐵𝑥 = 𝐶𝑥 = 𝐴))
21abbii 2726 . 2 {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)} = {𝑥 ∣ (𝑥 = 𝐵𝑥 = 𝐶𝑥 = 𝐴)}
3 dftp2 4178 . 2 {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}
4 dftp2 4178 . 2 {𝐵, 𝐶, 𝐴} = {𝑥 ∣ (𝑥 = 𝐵𝑥 = 𝐶𝑥 = 𝐴)}
52, 3, 43eqtr4i 2642 1 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
Colors of variables: wff setvar class
Syntax hints:  w3o 1030   = wceq 1475  {cab 2596  {ctp 4129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128  df-tp 4130
This theorem is referenced by:  tpcomb  4230  tpass  4231  tpidm13  4235  tpidm23  4236  tpnzd  4257  tpprceq3  4276  fvtp2  6366  fvtp3  6367  fvtp2g  6369  fvtp3g  6370  f13dfv  6430  en3lplem2  8395  estrres  16602  nb3graprlem2  25981  nb3grapr  25982  nb3grapr2  25983  nb3gra2nb  25984  cusgra3v  25993  frgra3v  26529  1to3vfriswmgra  26534  dvh4dimN  35754  en3lplem2VD  38101  nb3grprlem2  40609  nb3grpr  40610  nb3grpr2  40611  nb3gr2nb  40612  cplgr3v  40657  frgr3v  41445  1to3vfriswmgr  41450
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