Theorem un12 3130

Description: A rearrangement of union. (Contributed by NM, 12-Aug-2004.)

Assertion

Ref	Expression
un12	|- ( A u. ( B u. C ) ) = ( B u. ( A u. C ) )

Proof of Theorem un12

Step	Hyp	Ref	Expression
1		uncom 3116	. . 3 |- ( A u. B ) = ( B u. A )
2	1	uneq1i 3122	. 2 |- ( ( A u. B ) u. C ) = ( ( B u. A ) u. C )
3		unass 3129	. 2 |- ( ( A u. B ) u. C ) = ( A u. ( B u. C ) )
4		unass 3129	. 2 |- ( ( B u. A ) u. C ) = ( B u. ( A u. C ) )
5	2, 3, 4	3eqtr3i 2109	1 |- ( A u. ( B u. C ) ) = ( B u. ( A u. C ) )

Syntax hints:   = wceq 1284   u. cun 2971

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

This theorem is referenced by:  un23  3131  un4  3132