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Theorem indi 3291
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 790 . . . 4 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∨ (𝑥𝐴𝑥𝐶)))
2 elin 3227 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3 elin 3227 . . . . 5 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
42, 3orbi12i 736 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∨ (𝑥𝐴𝑥𝐶)))
51, 4bitr4i 186 . . 3 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)))
6 elun 3185 . . . 4 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
76anbi2i 450 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
8 elun 3185 . . 3 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)))
95, 7, 83bitr4i 211 . 2 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)))
109ineqri 3237 1 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  wa 103  wo 680   = wceq 1314  wcel 1463  cun 3037  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-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-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045
This theorem is referenced by:  indir  3293  undisj2  3389  disjssun  3394  difdifdirss  3415  disjpr2  3555  diftpsn3  3629  resundi  4800
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