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Theorem tprot 3586
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 953 . . 3  |-  ( ( x  =  A  \/  x  =  B  \/  x  =  C )  <->  ( x  =  B  \/  x  =  C  \/  x  =  A )
)
21abbii 2233 . 2  |-  { x  |  ( x  =  A  \/  x  =  B  \/  x  =  C ) }  =  { x  |  (
x  =  B  \/  x  =  C  \/  x  =  A ) }
3 dftp2 3542 . 2  |-  { A ,  B ,  C }  =  { x  |  ( x  =  A  \/  x  =  B  \/  x  =  C ) }
4 dftp2 3542 . 2  |-  { B ,  C ,  A }  =  { x  |  ( x  =  B  \/  x  =  C  \/  x  =  A ) }
52, 3, 43eqtr4i 2148 1  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
Colors of variables: wff set class
Syntax hints:    \/ w3o 946    = wceq 1316   {cab 2103   {ctp 3499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3or 948  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-tp 3505
This theorem is referenced by:  tpcomb  3588  tpass  3589  tpidm13  3593  tpidm23  3594  prsstp23  3645  fvtp2g  5597  fvtp3g  5598  fvtp2  5600  fvtp3  5601
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