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Theorem tprot 3687
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 984 . . 3 ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) ↔ (𝑥 = 𝐵𝑥 = 𝐶𝑥 = 𝐴))
21abbii 2293 . 2 {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)} = {𝑥 ∣ (𝑥 = 𝐵𝑥 = 𝐶𝑥 = 𝐴)}
3 dftp2 3643 . 2 {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}
4 dftp2 3643 . 2 {𝐵, 𝐶, 𝐴} = {𝑥 ∣ (𝑥 = 𝐵𝑥 = 𝐶𝑥 = 𝐴)}
52, 3, 43eqtr4i 2208 1 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
Colors of variables: wff set class
Syntax hints:  w3o 977   = wceq 1353  {cab 2163  {ctp 3596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3or 979  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-tp 3602
This theorem is referenced by:  tpcomb  3689  tpass  3690  tpidm13  3694  tpidm23  3695  prsstp23  3749  fvtp2g  5727  fvtp3g  5728  fvtp2  5730  fvtp3  5731
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