Theorem un23 3230

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 3228	. 2 |- ( ( A u. B ) u. C ) = ( A u. ( B u. C ) )
2		un12 3229	. 2 |- ( A u. ( B u. C ) ) = ( B u. ( A u. C ) )
3		uncom 3215	. 2 |- ( B u. ( A u. C ) ) = ( ( A u. C ) u. B )
4	1, 2, 3	3eqtri 2162	1 |- ( ( A u. B ) u. C ) = ( ( A u. C ) u. B )

Syntax hints:   = wceq 1331   u. cun 3064

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070

This theorem is referenced by:  setscom  11988