Theorem un12 3362

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 3348	. . 3 |- ( A u. B ) = ( B u. A )
2	1	uneq1i 3354	. 2 |- ( ( A u. B ) u. C ) = ( ( B u. A ) u. C )
3		unass 3361	. 2 |- ( ( A u. B ) u. C ) = ( A u. ( B u. C ) )
4		unass 3361	. 2 |- ( ( B u. A ) u. C ) = ( B u. ( A u. C ) )
5	2, 3, 4	3eqtr3i 2258	1 |- ( A u. ( B u. C ) ) = ( B u. ( A u. C ) )

Syntax hints:   = wceq 1395   u. cun 3195

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201

This theorem is referenced by:  un23  3363  un4  3364