MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  inindir Structured version   Visualization version   GIF version

Theorem inindir 4226
Description: Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
inindir ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Proof of Theorem inindir
StepHypRef Expression
1 inidm 4217 . . 3 (𝐶𝐶) = 𝐶
21ineq2i 4207 . 2 ((𝐴𝐵) ∩ (𝐶𝐶)) = ((𝐴𝐵) ∩ 𝐶)
3 in4 4224 . 2 ((𝐴𝐵) ∩ (𝐶𝐶)) = ((𝐴𝐶) ∩ (𝐵𝐶))
42, 3eqtr3i 2755 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cin 3943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-in 3951
This theorem is referenced by:  difindir  4281  resindir  6002  predin  6335  restbas  23106  connsuba  23368  kgentopon  23486  trfbas2  23791  trfil2  23835  fclsrest  23972  trust  24178  chtdif  27135  ppidif  27140  mdslmd1lem1  32207  mdslmd1lem2  32208  mddmdin0i  32313  ballotlemgun  34275  cvmsss2  35015
  Copyright terms: Public domain W3C validator