Theorem un23 3281

Description: A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
un23	|- ( ( A u. B ) u. C ) = ( ( A u. C ) u. B )

Proof of Theorem un23

Step	Hyp	Ref	Expression
1		unass 3279	. 2 |- ( ( A u. B ) u. C ) = ( A u. ( B u. C ) )
2		un12 3280	. 2 |- ( A u. ( B u. C ) ) = ( B u. ( A u. C ) )
3		uncom 3266	. 2 |- ( B u. ( A u. C ) ) = ( ( A u. C ) u. B )
4	1, 2, 3	3eqtri 2190	1 |- ( ( A u. B ) u. C ) = ( ( A u. C ) u. B )

Syntax hints:   = wceq 1343   u. cun 3114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120

This theorem is referenced by:  setscom  12434