Theorem tpcoma 3486

Description: Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.)

Assertion

Ref	Expression
tpcoma	|- { A , B , C } = { B , A , C }

Proof of Theorem tpcoma

Step	Hyp	Ref	Expression
1		prcom 3468	. . 3 |- { A , B } = { B , A }
2	1	uneq1i 3122	. 2 |- ( { A , B } u. { C } ) = ( { B , A } u. { C } )
3		df-tp 3406	. 2 |- { A , B , C } = ( { A , B } u. { C } )
4		df-tp 3406	. 2 |- { B , A , C } = ( { B , A } u. { C } )
5	2, 3, 4	3eqtr4i 2111	1 |- { A , B , C } = { B , A , C }

Syntax hints:   = wceq 1284   u. cun 2971   {csn 3398   {cpr 3399   {ctp 3400

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-pr 3405  df-tp 3406

This theorem is referenced by:  tpcomb  3487