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Theorem difundi 3332
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 3084 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 eldif 3084 . . . 4 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐶))
31, 2anbi12i 456 . . 3 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥 ∈ (𝐴𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
4 elin 3263 . . 3 (𝑥 ∈ ((𝐴𝐵) ∩ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝑥 ∈ (𝐴𝐶)))
5 eldif 3084 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐶)))
6 elun 3221 . . . . . . . 8 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
76notbii 658 . . . . . . 7 𝑥 ∈ (𝐵𝐶) ↔ ¬ (𝑥𝐵𝑥𝐶))
87anbi2i 453 . . . . . 6 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
95, 8bitri 183 . . . . 5 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
10 ioran 742 . . . . . 6 (¬ (𝑥𝐵𝑥𝐶) ↔ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶))
1110anbi2i 453 . . . . 5 ((𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶)))
129, 11bitri 183 . . . 4 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶)))
13 anandi 580 . . . 4 ((𝑥𝐴 ∧ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
1412, 13bitri 183 . . 3 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
153, 4, 143bitr4ri 212 . 2 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∩ (𝐴𝐶)))
1615eqriv 2137 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wo 698   = wceq 1332  wcel 1481  cdif 3072  cun 3073  cin 3074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3077  df-un 3079  df-in 3081
This theorem is referenced by:  undm  3338  undifdc  6819  uncld  12319
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