Theorem indir 4232

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 4230	. 2 ⊢ (𝐶 ∩ (𝐴 ∪ 𝐵)) = ((𝐶 ∩ 𝐴) ∪ (𝐶 ∩ 𝐵))
2		incom 4155	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴 ∪ 𝐵))
3		incom 4155	. . 3 ⊢ (𝐴 ∩ 𝐶) = (𝐶 ∩ 𝐴)
4		incom 4155	. . 3 ⊢ (𝐵 ∩ 𝐶) = (𝐶 ∩ 𝐵)
5	3, 4	uneq12i 4113	. 2 ⊢ ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶)) = ((𝐶 ∩ 𝐴) ∪ (𝐶 ∩ 𝐵))
6	1, 2, 5	3eqtr4i 2794	1 ⊢ ((𝐴 ∪ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶))

This proof depends on syntax axioms:   = wceq 1570   ∪ cun 3897   ∩ cin 3898

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-un 3904  df-in 3906

This theorem is used by:  difundir  4237  undisj1  4415  disjpr2  4674  resundir  5985  predun  6331  djuassen  10257  fin23lem26  10403  fpwwe2lem12  10727  neitr  23498  fiuncmp  23722  connsuba  23738  trfil2  24206  tsmsres  24463  trust  24548  restmetu  24889  volun  25866  uniioombllem3  25906  itgsplitioo  26158  ppiprm  27478  chtprm  27480  chtdif  27485  ppidif  27490  cycpmco2f1  33685  carsgclctunlem1  34949  ballotlemfp1  35124  ballotlemgun  35157  mrsubvrs  36287  mthmpps  36347  fixun  36671  mbfposadd  38586  iunrelexp0  44702  31prm  48682