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

Theorem inindir 3793
 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 3784 . . 3 (𝐶𝐶) = 𝐶
21ineq2i 3773 . 2 ((𝐴𝐵) ∩ (𝐶𝐶)) = ((𝐴𝐵) ∩ 𝐶)
3 in4 3791 . 2 ((𝐴𝐵) ∩ (𝐶𝐶)) = ((𝐴𝐶) ∩ (𝐵𝐶))
42, 3eqtr3i 2634 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∩ cin 3539 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547 This theorem is referenced by:  difindir  3841  resindir  5333  predin  5620  restbas  20772  consuba  21033  kgentopon  21151  trfbas2  21457  trfil2  21501  fclsrest  21638  trust  21843  chtdif  24684  ppidif  24689  mdslmd1lem1  28568  mdslmd1lem2  28569  mddmdin0i  28674  ballotlemgun  29913  cvmsss2  30510
 Copyright terms: Public domain W3C validator