Theorem tprot 3736

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 987	. . 3 |- ( ( x = A \/ x = B \/ x = C ) <-> ( x = B \/ x = C \/ x = A ) )
2	1	abbii 2323	. 2 |- { x | ( x = A \/ x = B \/ x = C ) } = { x | ( x = B \/ x = C \/ x = A ) }
3		dftp2 3692	. 2 |- { A , B , C } = { x | ( x = A \/ x = B \/ x = C ) }
4		dftp2 3692	. 2 |- { B , C , A } = { x | ( x = B \/ x = C \/ x = A ) }
5	2, 3, 4	3eqtr4i 2238	1 |- { A , B , C } = { B , C , A }

Syntax hints:   \/ w3o 980   = wceq 1373   {cab 2193   {ctp 3645

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3or 982  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-tp 3651

This theorem is referenced by:  tpcomb  3738  tpass  3739  tpidm13  3743  tpidm23  3744  prsstp23  3799  fvtp2g  5816  fvtp3g  5817  fvtp2  5819  fvtp3  5820