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Theorem tprot 3756
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 940 . . 3  |-  ( ( x  =  A  \/  x  =  B  \/  x  =  C )  <->  ( x  =  B  \/  x  =  C  \/  x  =  A )
)
21abbii 2428 . 2  |-  { x  |  ( x  =  A  \/  x  =  B  \/  x  =  C ) }  =  { x  |  (
x  =  B  \/  x  =  C  \/  x  =  A ) }
3 dftp2 3713 . 2  |-  { A ,  B ,  C }  =  { x  |  ( x  =  A  \/  x  =  B  \/  x  =  C ) }
4 dftp2 3713 . 2  |-  { B ,  C ,  A }  =  { x  |  ( x  =  B  \/  x  =  C  \/  x  =  A ) }
52, 3, 43eqtr4i 2346 1  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
Colors of variables: wff set class
Syntax hints:    \/ w3o 933    = wceq 1633   {cab 2302   {ctp 3676
This theorem is referenced by:  tpcomb  3758  tpass  3759  tpidm13  3763  tpidm23  3764  fvtp2  5764  fvtp3  5765  en3lplem2  7462  tpprceq3  27227  fvtp2g  27238  fvtp3g  27239  nb3graprlem2  27398  nb3grapr  27399  nb3grapr2  27400  nb3gra2nb  27401  cusgra3v  27409  frgra3v  27594  1to3vfriswmgra  27599  en3lplem2VD  28131  dvh4dimN  31455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824  df-un 3191  df-sn 3680  df-pr 3681  df-tp 3682
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