Theorem indir 3470

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 3468	. 2 |- ( C i^i ( A u. B ) ) = ( ( C i^i A ) u. ( C i^i B ) )
2		incom 3411	. 2 |- ( ( A u. B ) i^i C ) = ( C i^i ( A u. B ) )
3		incom 3411	. . 3 |- ( A i^i C ) = ( C i^i A )
4		incom 3411	. . 3 |- ( B i^i C ) = ( C i^i B )
5	3, 4	uneq12i 3371	. 2 |- ( ( A i^i C ) u. ( B i^i C ) ) = ( ( C i^i A ) u. ( C i^i B ) )
6	1, 2, 5	3eqtr4i 2263	1 |- ( ( A u. B ) i^i C ) = ( ( A i^i C ) u. ( B i^i C ) )

Syntax hints:   = wceq 1398   u. cun 3209   i^i cin 3210

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217

This theorem is referenced by:  difundir  3474  undisj1  3566  disjpr2  3753  resundir  5052  djuassen  7524