Theorem unundir 3321

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 3302	. . 3 |- ( C u. C ) = C
2	1	uneq2i 3310	. 2 |- ( ( A u. B ) u. ( C u. C ) ) = ( ( A u. B ) u. C )
3		un4 3319	. 2 |- ( ( A u. B ) u. ( C u. C ) ) = ( ( A u. C ) u. ( B u. C ) )
4	2, 3	eqtr3i 2216	1 |- ( ( A u. B ) u. C ) = ( ( A u. C ) u. ( B u. C ) )

Syntax hints:   = wceq 1364   u. cun 3151

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157

This theorem is referenced by: (None)