Theorem unundir 3243

Description: Union distributes over itself. (Contributed by NM, 17-Aug-2004.)

Assertion

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

Proof of Theorem unundir

Step	Hyp	Ref	Expression
1		unidm 3224	. . 3 |- ( C u. C ) = C
2	1	uneq2i 3232	. 2 |- ( ( A u. B ) u. ( C u. C ) ) = ( ( A u. B ) u. C )
3		un4 3241	. 2 |- ( ( A u. B ) u. ( C u. C ) ) = ( ( A u. C ) u. ( B u. C ) )
4	2, 3	eqtr3i 2163	1 |- ( ( A u. B ) u. C ) = ( ( A u. C ) u. ( B u. C ) )

Syntax hints:   = wceq 1332   u. cun 3074

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080

This theorem is referenced by: (None)