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Theorem indifdir 3255
Description: Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
Assertion
Ref Expression
indifdir ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))

Proof of Theorem indifdir
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elin 3183 . . . 4 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
2 elin 3183 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
32notbii 629 . . . 4 𝑥 ∈ (𝐵𝐶) ↔ ¬ (𝑥𝐵𝑥𝐶))
41, 3anbi12i 448 . . 3 ((𝑥 ∈ (𝐴𝐶) ∧ ¬ 𝑥 ∈ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
5 eldif 3008 . . 3 (𝑥 ∈ ((𝐴𝐶) ∖ (𝐵𝐶)) ↔ (𝑥 ∈ (𝐴𝐶) ∧ ¬ 𝑥 ∈ (𝐵𝐶)))
6 elin 3183 . . . . 5 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶))
7 eldif 3008 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
87anbi1i 446 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶))
96, 8bitri 182 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶))
10 an32 529 . . . . 5 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵))
11 simpl 107 . . . . . . . 8 ((𝑥𝐵𝑥𝐶) → 𝑥𝐵)
1211con3i 597 . . . . . . 7 𝑥𝐵 → ¬ (𝑥𝐵𝑥𝐶))
1312anim2i 334 . . . . . 6 (((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵) → ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
14 simpl 107 . . . . . . 7 (((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)) → (𝑥𝐴𝑥𝐶))
15 ax-in2 580 . . . . . . . . . . 11 (¬ (𝑥𝐵𝑥𝐶) → ((𝑥𝐵𝑥𝐶) → ⊥))
1615expcomd 1375 . . . . . . . . . 10 (¬ (𝑥𝐵𝑥𝐶) → (𝑥𝐶 → (𝑥𝐵 → ⊥)))
1716impcom 123 . . . . . . . . 9 ((𝑥𝐶 ∧ ¬ (𝑥𝐵𝑥𝐶)) → (𝑥𝐵 → ⊥))
18 dfnot 1307 . . . . . . . . 9 𝑥𝐵 ↔ (𝑥𝐵 → ⊥))
1917, 18sylibr 132 . . . . . . . 8 ((𝑥𝐶 ∧ ¬ (𝑥𝐵𝑥𝐶)) → ¬ 𝑥𝐵)
2019adantll 460 . . . . . . 7 (((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)) → ¬ 𝑥𝐵)
2114, 20jca 300 . . . . . 6 (((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)) → ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵))
2213, 21impbii 124 . . . . 5 (((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
2310, 22bitri 182 . . . 4 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
249, 23bitri 182 . . 3 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
254, 5, 243bitr4ri 211 . 2 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ 𝑥 ∈ ((𝐴𝐶) ∖ (𝐵𝐶)))
2625eqriv 2085 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102   = wceq 1289  wfal 1294  wcel 1438  cdif 2996  cin 2998
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-in 3005
This theorem is referenced by: (None)
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