Theorem indir 4249

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 4247	. 2 ⊢ (𝐶 ∩ (𝐴 ∪ 𝐵)) = ((𝐶 ∩ 𝐴) ∪ (𝐶 ∩ 𝐵))
2		incom 4172	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴 ∪ 𝐵))
3		incom 4172	. . 3 ⊢ (𝐴 ∩ 𝐶) = (𝐶 ∩ 𝐴)
4		incom 4172	. . 3 ⊢ (𝐵 ∩ 𝐶) = (𝐶 ∩ 𝐵)
5	3, 4	uneq12i 4129	. 2 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶)) = ((𝐶 ∩ 𝐴) ∪ (𝐶 ∩ 𝐵))
6	1, 2, 5	3eqtr4i 2762	1 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶))

Syntax hints:   = wceq 1540   ∪ cun 3912   ∩ cin 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-un 3919  df-in 3921

This theorem is referenced by:  difundir  4254  undisj1  4425  disjpr2  4677  resundir  5965  predun  6301  djuassen  10132  fin23lem26  10278  fpwwe2lem12  10595  neitr  23067  fiuncmp  23291  connsuba  23307  trfil2  23774  tsmsres  24031  trust  24117  restmetu  24458  volun  25446  uniioombllem3  25486  itgsplitioo  25739  ppiprm  27061  chtprm  27063  chtdif  27068  ppidif  27073  cycpmco2f1  33081  carsgclctunlem1  34308  ballotlemfp1  34483  ballotlemgun  34516  mrsubvrs  35509  mthmpps  35569  fixun  35897  mbfposadd  37661  iunrelexp0  43691  31prm  47598