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Theorem difundi 3232
Description: Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
difundi (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Proof of Theorem difundi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eldif 2991 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 eldif 2991 . . . 4 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐶))
31, 2anbi12i 448 . . 3 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥 ∈ (𝐴𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
4 elin 3165 . . 3 (𝑥 ∈ ((𝐴𝐵) ∩ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝑥 ∈ (𝐴𝐶)))
5 eldif 2991 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐶)))
6 elun 3123 . . . . . . . 8 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
76notbii 627 . . . . . . 7 𝑥 ∈ (𝐵𝐶) ↔ ¬ (𝑥𝐵𝑥𝐶))
87anbi2i 445 . . . . . 6 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
95, 8bitri 182 . . . . 5 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
10 ioran 702 . . . . . 6 (¬ (𝑥𝐵𝑥𝐶) ↔ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶))
1110anbi2i 445 . . . . 5 ((𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶)))
129, 11bitri 182 . . . 4 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶)))
13 anandi 555 . . . 4 ((𝑥𝐴 ∧ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
1412, 13bitri 182 . . 3 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
153, 4, 143bitr4ri 211 . 2 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∩ (𝐴𝐶)))
1615eqriv 2080 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102  wo 662   = wceq 1285  wcel 1434  cdif 2979  cun 2980  cin 2981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-dif 2984  df-un 2986  df-in 2988
This theorem is referenced by:  undm  3238  undifdc  6468
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