Theorem tpcoma 3687

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 3669	. . 3 |- { A , B } = { B , A }
2	1	uneq1i 3286	. 2 |- ( { A , B } u. { C } ) = ( { B , A } u. { C } )
3		df-tp 3601	. 2 |- { A , B , C } = ( { A , B } u. { C } )
4		df-tp 3601	. 2 |- { B , A , C } = ( { B , A } u. { C } )
5	2, 3, 4	3eqtr4i 2208	1 |- { A , B , C } = { B , A , C }

Syntax hints:   = wceq 1353   u. cun 3128   {csn 3593   {cpr 3594   {ctp 3595

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-pr 3600  df-tp 3601

This theorem is referenced by:  tpcomb  3688