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

Theorem difindir 4242
 Description: Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
difindir ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Proof of Theorem difindir
StepHypRef Expression
1 inindir 4187 . 2 ((𝐴𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ (V ∖ 𝐶)) ∩ (𝐵 ∩ (V ∖ 𝐶)))
2 invdif 4228 . 2 ((𝐴𝐵) ∩ (V ∖ 𝐶)) = ((𝐴𝐵) ∖ 𝐶)
3 invdif 4228 . . 3 (𝐴 ∩ (V ∖ 𝐶)) = (𝐴𝐶)
4 invdif 4228 . . 3 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
53, 4ineq12i 4170 . 2 ((𝐴 ∩ (V ∖ 𝐶)) ∩ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴𝐶) ∩ (𝐵𝐶))
61, 2, 53eqtr3i 2855 1 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  Vcvv 3479   ∖ cdif 3915   ∩ cin 3917 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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-rab 3141  df-v 3481  df-dif 3921  df-in 3925 This theorem is referenced by:  ablfac1eulem  19185  ballotlemgun  31802
 Copyright terms: Public domain W3C validator