Theorem undir 3372

Description: Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)

Assertion

Ref	Expression
undir	|- ( ( A i^i B ) u. C ) = ( ( A u. C ) i^i ( B u. C ) )

Proof of Theorem undir

Step	Hyp	Ref	Expression
1		undi 3370	. 2 |- ( C u. ( A i^i B ) ) = ( ( C u. A ) i^i ( C u. B ) )
2		uncom 3266	. 2 |- ( ( A i^i B ) u. C ) = ( C u. ( A i^i B ) )
3		uncom 3266	. . 3 |- ( A u. C ) = ( C u. A )
4		uncom 3266	. . 3 |- ( B u. C ) = ( C u. B )
5	3, 4	ineq12i 3321	. 2 |- ( ( A u. C ) i^i ( B u. C ) ) = ( ( C u. A ) i^i ( C u. B ) )
6	1, 2, 5	3eqtr4i 2196	1 |- ( ( A i^i B ) u. C ) = ( ( A u. C ) i^i ( B u. C ) )

Syntax hints:   = wceq 1343   u. cun 3114   i^i cin 3115

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 2297  df-v 2728  df-un 3120  df-in 3122

This theorem is referenced by: (None)