Theorem un12 3367

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 3353	. . 3 |- ( A u. B ) = ( B u. A )
2	1	uneq1i 3359	. 2 |- ( ( A u. B ) u. C ) = ( ( B u. A ) u. C )
3		unass 3366	. 2 |- ( ( A u. B ) u. C ) = ( A u. ( B u. C ) )
4		unass 3366	. 2 |- ( ( B u. A ) u. C ) = ( B u. ( A u. C ) )
5	2, 3, 4	3eqtr3i 2260	1 |- ( A u. ( B u. C ) ) = ( B u. ( A u. C ) )

Syntax hints:   = wceq 1398   u. cun 3199

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205

This theorem is referenced by:  un23  3368  un4  3369