Theorem tprot 3764

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 1010	. . 3 |- ( ( x = A \/ x = B \/ x = C ) <-> ( x = B \/ x = C \/ x = A ) )
2	1	abbii 2347	. 2 |- { x | ( x = A \/ x = B \/ x = C ) } = { x | ( x = B \/ x = C \/ x = A ) }
3		dftp2 3718	. 2 |- { A , B , C } = { x | ( x = A \/ x = B \/ x = C ) }
4		dftp2 3718	. 2 |- { B , C , A } = { x | ( x = B \/ x = C \/ x = A ) }
5	2, 3, 4	3eqtr4i 2262	1 |- { A , B , C } = { B , C , A }

Syntax hints:   \/ w3o 1003   = wceq 1397   {cab 2217   {ctp 3671

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-tp 3677

This theorem is referenced by:  tpcomb  3766  tpass  3767  tpidm13  3771  tpidm23  3772  prsstp23  3828  fvtp2g  5862  fvtp3g  5863  fvtp2  5865  fvtp3  5866