Theorem tprot 3648

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.

Step	Hyp	Ref	Expression
1		3orrot 969	. . 3 |- ( ( x = A \/ x = B \/ x = C ) <-> ( x = B \/ x = C \/ x = A ) )
2	1	abbii 2270	. 2 |- { x | ( x = A \/ x = B \/ x = C ) } = { x | ( x = B \/ x = C \/ x = A ) }
3		dftp2 3604	. 2 |- { A , B , C } = { x | ( x = A \/ x = B \/ x = C ) }
4		dftp2 3604	. 2 |- { B , C , A } = { x | ( x = B \/ x = C \/ x = A ) }
5	2, 3, 4	3eqtr4i 2185	1 |- { A , B , C } = { B , C , A }

Syntax hints:   \/ w3o 962   = wceq 1332   {cab 2140   {ctp 3558

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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-3or 964  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-un 3102  df-sn 3562  df-pr 3563  df-tp 3564

This theorem is referenced by:  tpcomb  3650  tpass  3651  tpidm13  3655  tpidm23  3656  prsstp23  3707  fvtp2g  5669  fvtp3g  5670  fvtp2  5672  fvtp3  5673