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Theorem difundi 3186
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 2924 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 eldif 2924 . . . 4 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐶))
31, 2anbi12i 433 . . 3 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥 ∈ (𝐴𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
4 elin 3123 . . 3 (𝑥 ∈ ((𝐴𝐵) ∩ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝑥 ∈ (𝐴𝐶)))
5 eldif 2924 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐶)))
6 elun 3081 . . . . . . . 8 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
76notbii 594 . . . . . . 7 𝑥 ∈ (𝐵𝐶) ↔ ¬ (𝑥𝐵𝑥𝐶))
87anbi2i 430 . . . . . 6 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
95, 8bitri 173 . . . . 5 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)))
10 ioran 669 . . . . . 6 (¬ (𝑥𝐵𝑥𝐶) ↔ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶))
1110anbi2i 430 . . . . 5 ((𝑥𝐴 ∧ ¬ (𝑥𝐵𝑥𝐶)) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶)))
129, 11bitri 173 . . . 4 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶)))
13 anandi 524 . . . 4 ((𝑥𝐴 ∧ (¬ 𝑥𝐵 ∧ ¬ 𝑥𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
1412, 13bitri 173 . . 3 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ (𝑥𝐴 ∧ ¬ 𝑥𝐶)))
153, 4, 143bitr4ri 202 . 2 (𝑥 ∈ (𝐴 ∖ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∩ (𝐴𝐶)))
1615eqriv 2037 1 (𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 97  wo 629   = wceq 1243  wcel 1393  cdif 2911  cun 2912  cin 2913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-dif 2917  df-un 2919  df-in 2921
This theorem is referenced by:  undm  3192
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