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Theorem indifdir 3300
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 3227 . . . 4 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
2 elin 3227 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
32notbii 640 . . . 4 𝑥 ∈ (𝐵𝐶) ↔ ¬ (𝑥𝐵𝑥𝐶))
41, 3anbi12i 453 . . 3 ((𝑥 ∈ (𝐴𝐶) ∧ ¬ 𝑥 ∈ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
5 eldif 3048 . . 3 (𝑥 ∈ ((𝐴𝐶) ∖ (𝐵𝐶)) ↔ (𝑥 ∈ (𝐴𝐶) ∧ ¬ 𝑥 ∈ (𝐵𝐶)))
6 elin 3227 . . . . 5 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶))
7 eldif 3048 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
87anbi1i 451 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶))
96, 8bitri 183 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶))
10 an32 534 . . . . 5 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵))
11 simpl 108 . . . . . . . 8 ((𝑥𝐵𝑥𝐶) → 𝑥𝐵)
1211con3i 604 . . . . . . 7 𝑥𝐵 → ¬ (𝑥𝐵𝑥𝐶))
1312anim2i 337 . . . . . 6 (((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵) → ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
14 simpl 108 . . . . . . 7 (((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)) → (𝑥𝐴𝑥𝐶))
15 ax-in2 587 . . . . . . . . . . 11 (¬ (𝑥𝐵𝑥𝐶) → ((𝑥𝐵𝑥𝐶) → ⊥))
1615expcomd 1400 . . . . . . . . . 10 (¬ (𝑥𝐵𝑥𝐶) → (𝑥𝐶 → (𝑥𝐵 → ⊥)))
1716impcom 124 . . . . . . . . 9 ((𝑥𝐶 ∧ ¬ (𝑥𝐵𝑥𝐶)) → (𝑥𝐵 → ⊥))
18 dfnot 1332 . . . . . . . . 9 𝑥𝐵 ↔ (𝑥𝐵 → ⊥))
1917, 18sylibr 133 . . . . . . . 8 ((𝑥𝐶 ∧ ¬ (𝑥𝐵𝑥𝐶)) → ¬ 𝑥𝐵)
2019adantll 465 . . . . . . 7 (((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)) → ¬ 𝑥𝐵)
2114, 20jca 302 . . . . . 6 (((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)) → ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵))
2213, 21impbii 125 . . . . 5 (((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
2310, 22bitri 183 . . . 4 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
249, 23bitri 183 . . 3 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
254, 5, 243bitr4ri 212 . 2 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ 𝑥 ∈ ((𝐴𝐶) ∖ (𝐵𝐶)))
2625eqriv 2112 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1314  wfal 1319  wcel 1463  cdif 3036  cin 3038
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-in 3045
This theorem is referenced by: (None)
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