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Theorem indifdir 3406
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 3333 . . . 4 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
2 elin 3333 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
32notbii 669 . . . 4 𝑥 ∈ (𝐵𝐶) ↔ ¬ (𝑥𝐵𝑥𝐶))
41, 3anbi12i 460 . . 3 ((𝑥 ∈ (𝐴𝐶) ∧ ¬ 𝑥 ∈ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
5 eldif 3153 . . 3 (𝑥 ∈ ((𝐴𝐶) ∖ (𝐵𝐶)) ↔ (𝑥 ∈ (𝐴𝐶) ∧ ¬ 𝑥 ∈ (𝐵𝐶)))
6 elin 3333 . . . . 5 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶))
7 eldif 3153 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
87anbi1i 458 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶))
96, 8bitri 184 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶))
10 an32 562 . . . . 5 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵))
11 simpl 109 . . . . . . . 8 ((𝑥𝐵𝑥𝐶) → 𝑥𝐵)
1211con3i 633 . . . . . . 7 𝑥𝐵 → ¬ (𝑥𝐵𝑥𝐶))
1312anim2i 342 . . . . . 6 (((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵) → ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
14 simpl 109 . . . . . . 7 (((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)) → (𝑥𝐴𝑥𝐶))
15 ax-in2 616 . . . . . . . . . . 11 (¬ (𝑥𝐵𝑥𝐶) → ((𝑥𝐵𝑥𝐶) → ⊥))
1615expcomd 1452 . . . . . . . . . 10 (¬ (𝑥𝐵𝑥𝐶) → (𝑥𝐶 → (𝑥𝐵 → ⊥)))
1716impcom 125 . . . . . . . . 9 ((𝑥𝐶 ∧ ¬ (𝑥𝐵𝑥𝐶)) → (𝑥𝐵 → ⊥))
18 dfnot 1382 . . . . . . . . 9 𝑥𝐵 ↔ (𝑥𝐵 → ⊥))
1917, 18sylibr 134 . . . . . . . 8 ((𝑥𝐶 ∧ ¬ (𝑥𝐵𝑥𝐶)) → ¬ 𝑥𝐵)
2019adantll 476 . . . . . . 7 (((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)) → ¬ 𝑥𝐵)
2114, 20jca 306 . . . . . 6 (((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)) → ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵))
2213, 21impbii 126 . . . . 5 (((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
2310, 22bitri 184 . . . 4 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
249, 23bitri 184 . . 3 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
254, 5, 243bitr4ri 213 . 2 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ 𝑥 ∈ ((𝐴𝐶) ∖ (𝐵𝐶)))
2625eqriv 2186 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104   = wceq 1364  wfal 1369  wcel 2160  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-fal 1370  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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