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Theorem undi 3850
Description: Distributive law for union over intersection. Exercise 11 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
undi (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Proof of Theorem undi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elin 3774 . . . 4 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
21orbi2i 541 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∨ (𝑥𝐵𝑥𝐶)))
3 ordi 907 . . 3 ((𝑥𝐴 ∨ (𝑥𝐵𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴𝑥𝐶)))
4 elin 3774 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∩ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝑥 ∈ (𝐴𝐶)))
5 elun 3731 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
6 elun 3731 . . . . 5 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
75, 6anbi12i 732 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥 ∈ (𝐴𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴𝑥𝐶)))
84, 7bitr2i 265 . . 3 (((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴𝑥𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∩ (𝐴𝐶)))
92, 3, 83bitri 286 . 2 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∩ (𝐴𝐶)))
109uneqri 3733 1 (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wo 383  wa 384   = wceq 1480  wcel 1987  cun 3553  cin 3554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-un 3560  df-in 3562
This theorem is referenced by:  undir  3852  dfif4  4073  dfif5  4074
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