Theorem indir 4227

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 4225	. 2 ⊢ (𝐶 ∩ (𝐴 ∪ 𝐵)) = ((𝐶 ∩ 𝐴) ∪ (𝐶 ∩ 𝐵))
2		incom 4150	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴 ∪ 𝐵))
3		incom 4150	. . 3 ⊢ (𝐴 ∩ 𝐶) = (𝐶 ∩ 𝐴)
4		incom 4150	. . 3 ⊢ (𝐵 ∩ 𝐶) = (𝐶 ∩ 𝐵)
5	3, 4	uneq12i 4107	. 2 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶)) = ((𝐶 ∩ 𝐴) ∪ (𝐶 ∩ 𝐵))
6	1, 2, 5	3eqtr4i 2770	1 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶))

Syntax hints:   = wceq 1542   ∪ cun 3888   ∩ cin 3889

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-un 3895  df-in 3897

This theorem is referenced by:  difundir  4232  undisj1  4403  disjpr2  4658  resundir  5955  predun  6288  djuassen  10096  fin23lem26  10242  fpwwe2lem12  10560  neitr  23159  fiuncmp  23383  connsuba  23399  trfil2  23866  tsmsres  24123  trust  24208  restmetu  24549  volun  25526  uniioombllem3  25566  itgsplitioo  25819  ppiprm  27132  chtprm  27134  chtdif  27139  ppidif  27144  cycpmco2f1  33204  carsgclctunlem1  34481  ballotlemfp1  34656  ballotlemgun  34689  mrsubvrs  35724  mthmpps  35784  fixun  36109  mbfposadd  38006  iunrelexp0  44151  31prm  48076