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Theorem difindir 3405
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 3368 . 2 ((𝐴𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ (V ∖ 𝐶)) ∩ (𝐵 ∩ (V ∖ 𝐶)))
2 invdif 3392 . 2 ((𝐴𝐵) ∩ (V ∖ 𝐶)) = ((𝐴𝐵) ∖ 𝐶)
3 invdif 3392 . . 3 (𝐴 ∩ (V ∖ 𝐶)) = (𝐴𝐶)
4 invdif 3392 . . 3 (𝐵 ∩ (V ∖ 𝐶)) = (𝐵𝐶)
53, 4ineq12i 3349 . 2 ((𝐴 ∩ (V ∖ 𝐶)) ∩ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴𝐶) ∩ (𝐵𝐶))
61, 2, 53eqtr3i 2218 1 ((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1364  Vcvv 2752  cdif 3141  cin 3143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-in 3150
This theorem is referenced by: (None)
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