Theorem tpcoma 4285

Description: Swap 1st and 2nd members of an unordered 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 4267	. . 3 |- { A , B } = { B , A }
2	1	uneq1i 3763	. 2 |- ( { A , B } u. { C } ) = ( { B , A } u. { C } )
3		df-tp 4182	. 2 |- { A , B , C } = ( { A , B } u. { C } )
4		df-tp 4182	. 2 |- { B , A , C } = ( { B , A } u. { C } )
5	2, 3, 4	3eqtr4i 2654	1 |- { A , B , C } = { B , A , C }

Syntax hints:   = wceq 1483   u. cun 3572   {csn 4177   {cpr 4179   {ctp 4181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-pr 4180  df-tp 4182

This theorem is referenced by:  tpcomb  4286  tppreqb  4336  nb3grpr2  26285  nb3gr2nb  26286  frgr3v  27139  3vfriswmgr  27142  1to3vfriswmgr  27144