Theorem tprot 3725

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 986	. . 3 |- ( ( x = A \/ x = B \/ x = C ) <-> ( x = B \/ x = C \/ x = A ) )
2	1	abbii 2320	. 2 |- { x | ( x = A \/ x = B \/ x = C ) } = { x | ( x = B \/ x = C \/ x = A ) }
3		dftp2 3681	. 2 |- { A , B , C } = { x | ( x = A \/ x = B \/ x = C ) }
4		dftp2 3681	. 2 |- { B , C , A } = { x | ( x = B \/ x = C \/ x = A ) }
5	2, 3, 4	3eqtr4i 2235	1 |- { A , B , C } = { B , C , A }

Syntax hints:   \/ w3o 979   = wceq 1372   {cab 2190   {ctp 3634

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3or 981  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-tp 3640

This theorem is referenced by:  tpcomb  3727  tpass  3728  tpidm13  3732  tpidm23  3733  prsstp23  3787  fvtp2g  5792  fvtp3g  5793  fvtp2  5795  fvtp3  5796