Theorem indir 4291

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

Assertion

Ref	Expression
indir	⊢ ((𝐴 ∪ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶))

Proof of Theorem indir

Step	Hyp	Ref	Expression
1		indi 4289	. 2 ⊢ (𝐶 ∩ (𝐴 ∪ 𝐵)) = ((𝐶 ∩ 𝐴) ∪ (𝐶 ∩ 𝐵))
2		incom 4216	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴 ∪ 𝐵))
3		incom 4216	. . 3 ⊢ (𝐴 ∩ 𝐶) = (𝐶 ∩ 𝐴)
4		incom 4216	. . 3 ⊢ (𝐵 ∩ 𝐶) = (𝐶 ∩ 𝐵)
5	3, 4	uneq12i 4175	. 2 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶)) = ((𝐶 ∩ 𝐴) ∪ (𝐶 ∩ 𝐵))
6	1, 2, 5	3eqtr4i 2772	1 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶))

Syntax hints:   = wceq 1536   ∪ cun 3960   ∩ cin 3961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-un 3967  df-in 3969

This theorem is referenced by:  difundir  4296  undisj1  4467  disjpr2  4717  resundir  6014  predun  6350  djuassen  10216  fin23lem26  10362  fpwwe2lem12  10679  neitr  23203  fiuncmp  23427  connsuba  23443  trfil2  23910  tsmsres  24167  trust  24253  restmetu  24598  volun  25593  uniioombllem3  25633  itgsplitioo  25887  ppiprm  27208  chtprm  27210  chtdif  27215  ppidif  27220  cycpmco2f1  33126  carsgclctunlem1  34298  ballotlemfp1  34472  ballotlemgun  34505  mrsubvrs  35506  mthmpps  35566  fixun  35890  mbfposadd  37653  metakunt17  42202  metakunt21  42206  metakunt22  42207  metakunt24  42209  iunrelexp0  43691  31prm  47521