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

Theorem indifdir 4185
Description: Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.) (Revised by BTernaryTau, 14-Aug-2024.)
Assertion
Ref Expression
indifdir ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))

Proof of Theorem indifdir
StepHypRef Expression
1 indifdi 4184 . 2 (𝐶 ∩ (𝐴𝐵)) = ((𝐶𝐴) ∖ (𝐶𝐵))
2 incom 4101 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴𝐵))
3 incom 4101 . . 3 (𝐴𝐶) = (𝐶𝐴)
4 incom 4101 . . 3 (𝐵𝐶) = (𝐶𝐵)
53, 4difeq12i 4021 . 2 ((𝐴𝐶) ∖ (𝐵𝐶)) = ((𝐶𝐴) ∖ (𝐶𝐵))
61, 2, 53eqtr4i 2772 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cdif 3850  cin 3852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-rab 3063  df-v 3402  df-dif 3856  df-in 3860
This theorem is referenced by:  resdifdir  6079  preddif  6164  fresaun  6559  uniioombllem4  24350  subsalsal  43480
  Copyright terms: Public domain W3C validator