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Theorem tprot 3620
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 969 . . 3 ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) ↔ (𝑥 = 𝐵𝑥 = 𝐶𝑥 = 𝐴))
21abbii 2256 . 2 {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)} = {𝑥 ∣ (𝑥 = 𝐵𝑥 = 𝐶𝑥 = 𝐴)}
3 dftp2 3576 . 2 {𝐴, 𝐵, 𝐶} = {𝑥 ∣ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶)}
4 dftp2 3576 . 2 {𝐵, 𝐶, 𝐴} = {𝑥 ∣ (𝑥 = 𝐵𝑥 = 𝐶𝑥 = 𝐴)}
52, 3, 43eqtr4i 2171 1 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
Colors of variables: wff set class
Syntax hints:  w3o 962   = wceq 1332  {cab 2126  {ctp 3530
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3or 964  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2689  df-un 3076  df-sn 3534  df-pr 3535  df-tp 3536
This theorem is referenced by:  tpcomb  3622  tpass  3623  tpidm13  3627  tpidm23  3628  prsstp23  3679  fvtp2g  5633  fvtp3g  5634  fvtp2  5636  fvtp3  5637
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