Theorem indir 3480

Description: Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)

Assertion

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

Proof of Theorem indir

Step	Hyp	Ref	Expression
1		indi 3478	. 2 |- ( C i^i ( A u. B ) ) = ( ( C i^i A ) u. ( C i^i B ) )
2		incom 3421	. 2 |- ( ( A u. B ) i^i C ) = ( C i^i ( A u. B ) )
3		incom 3421	. . 3 |- ( A i^i C ) = ( C i^i A )
4		incom 3421	. . 3 |- ( B i^i C ) = ( C i^i B )
5	3, 4	uneq12i 3381	. 2 |- ( ( A i^i C ) u. ( B i^i C ) ) = ( ( C i^i A ) u. ( C i^i B ) )
6	1, 2, 5	3eqtr4i 2269	1 |- ( ( A u. B ) i^i C ) = ( ( A i^i C ) u. ( B i^i C ) )

Syntax hints:   = wceq 1402   u. cun 3218   i^i cin 3219

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226

This theorem is referenced by:  difundir  3484  undisj1  3581  disjpr2  3769  resundir  5072  djuassen  7563  ballotfilemfp1  13209  ballotfilemgun  13246