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Theorem indi 3181
Description: Distributive law for intersection over union. Exercise 10 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
indi (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Proof of Theorem indi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 andi 731 . . . 4 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∨ (𝑥𝐴𝑥𝐶)))
2 elin 3123 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3 elin 3123 . . . . 5 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
42, 3orbi12i 681 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∨ (𝑥𝐴𝑥𝐶)))
51, 4bitr4i 176 . . 3 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)))
6 elun 3081 . . . 4 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
76anbi2i 430 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
8 elun 3081 . . 3 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)))
95, 7, 83bitr4i 201 . 2 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)))
109ineqri 3127 1 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  wa 97  wo 629   = wceq 1243  wcel 1393  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-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-un 2919  df-in 2921
This theorem is referenced by:  indir  3183  undisj2  3277  disjssun  3282  difdifdirss  3304  disjpr2  3431  diftpsn3  3502  resundi  4612
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