Theorem indir 4202

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 4200	. 2 ⊢ (𝐶 ∩ (𝐴 ∪ 𝐵)) = ((𝐶 ∩ 𝐴) ∪ (𝐶 ∩ 𝐵))
2		incom 4128	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴 ∪ 𝐵))
3		incom 4128	. . 3 ⊢ (𝐴 ∩ 𝐶) = (𝐶 ∩ 𝐴)
4		incom 4128	. . 3 ⊢ (𝐵 ∩ 𝐶) = (𝐶 ∩ 𝐵)
5	3, 4	uneq12i 4088	. 2 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶)) = ((𝐶 ∩ 𝐴) ∪ (𝐶 ∩ 𝐵))
6	1, 2, 5	3eqtr4i 2831	1 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶))

Syntax hints:   = wceq 1538   ∪ cun 3879   ∩ cin 3880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-un 3886  df-in 3888

This theorem is referenced by:  difundir  4207  undisj1  4369  disjpr2  4609  resundir  5833  predun  6140  djuassen  9589  fin23lem26  9736  fpwwe2lem13  10053  neitr  21785  fiuncmp  22009  connsuba  22025  trfil2  22492  tsmsres  22749  trust  22835  restmetu  23177  volun  24149  uniioombllem3  24189  itgsplitioo  24441  ppiprm  25736  chtprm  25738  chtdif  25743  ppidif  25748  cycpmco2f1  30816  carsgclctunlem1  31685  ballotlemfp1  31859  ballotlemgun  31892  mrsubvrs  32882  mthmpps  32942  fixun  33483  mbfposadd  35104  metakunt17  39366  metakunt21  39370  metakunt22  39371  metakunt24  39373  iunrelexp0  40403  31prm  44114