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Theorem difindir 4299
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 4244 . 2 ((𝐴𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ (V ∖ 𝐶)) ∩ (𝐵 ∩ (V ∖ 𝐶)))
2 invdif 4285 . 2 ((𝐴𝐵) ∩ (V ∖ 𝐶)) = ((𝐴𝐵) ∖ 𝐶)
3 invdif 4285 . . 3 (𝐴 ∩ (V ∖ 𝐶)) = (𝐴𝐶)
4 invdif 4285 . . 3 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
53, 4ineq12i 4226 . 2 ((𝐴 ∩ (V ∖ 𝐶)) ∩ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴𝐶) ∩ (𝐵𝐶))
61, 2, 53eqtr3i 2771 1 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  Vcvv 3478  cdif 3960  cin 3962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970
This theorem is referenced by:  ablfac1eulem  20107  ballotlemgun  34506
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