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Theorem indifdir 4218
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 4217 . 2 (𝐶 ∩ (𝐴𝐵)) = ((𝐶𝐴) ∖ (𝐶𝐵))
2 incom 4135 . 2 ((𝐴𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴𝐵))
3 incom 4135 . . 3 (𝐴𝐶) = (𝐶𝐴)
4 incom 4135 . . 3 (𝐵𝐶) = (𝐶𝐵)
53, 4difeq12i 4055 . 2 ((𝐴𝐶) ∖ (𝐵𝐶)) = ((𝐶𝐴) ∖ (𝐶𝐵))
61, 2, 53eqtr4i 2776 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  cdif 3884  cin 3886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-in 3894
This theorem is referenced by:  resdifdir  6140  preddif  6232  fresaun  6645  uniioombllem4  24750  subsalsal  43898
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