Theorem indir 3875

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 3873	. 2 |- ( C i^i ( A u. B ) ) = ( ( C i^i A ) u. ( C i^i B ) )
2		incom 3805	. 2 |- ( ( A u. B ) i^i C ) = ( C i^i ( A u. B ) )
3		incom 3805	. . 3 |- ( A i^i C ) = ( C i^i A )
4		incom 3805	. . 3 |- ( B i^i C ) = ( C i^i B )
5	3, 4	uneq12i 3765	. 2 |- ( ( A i^i C ) u. ( B i^i C ) ) = ( ( C i^i A ) u. ( C i^i B ) )
6	1, 2, 5	3eqtr4i 2654	1 |- ( ( A u. B ) i^i C ) = ( ( A i^i C ) u. ( B i^i C ) )

Syntax hints:   = wceq 1483   u. cun 3572   i^i cin 3573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581

This theorem is referenced by:  difundir  3880  undisj1  4029  disjpr2  4248  disjpr2OLD  4249  resundir  5411  predun  5704  cdaassen  9004  fin23lem26  9147  fpwwe2lem13  9464  neitr  20984  fiuncmp  21207  connsuba  21223  trfil2  21691  tsmsres  21947  trust  22033  restmetu  22375  volun  23313  uniioombllem3  23353  itgsplitioo  23604  ppiprm  24877  chtprm  24879  chtdif  24884  ppidif  24889  carsgclctunlem1  30379  ballotlemfp1  30553  ballotlemgun  30586  mrsubvrs  31419  mthmpps  31479  fixun  32016  mbfposadd  33457  iunrelexp0  37994  31prm  41512