Theorem unundi 3282

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

Assertion

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

Proof of Theorem unundi

Step	Hyp	Ref	Expression
1		unidm 3264	. . 3 |- ( A u. A ) = A
2	1	uneq1i 3271	. 2 |- ( ( A u. A ) u. ( B u. C ) ) = ( A u. ( B u. C ) )
3		un4 3281	. 2 |- ( ( A u. A ) u. ( B u. C ) ) = ( ( A u. B ) u. ( A u. C ) )
4	2, 3	eqtr3i 2188	1 |- ( A u. ( B u. C ) ) = ( ( A u. B ) u. ( A u. C ) )

Syntax hints:   = wceq 1343   u. cun 3113

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727  df-un 3119

This theorem is referenced by:  unfiin  6887