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Theorem iunin1 4986
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4974 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
iunin1 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem iunin1
StepHypRef Expression
1 iunin2 4985 . 2 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
2 incom 4177 . . . 4 (𝐶𝐵) = (𝐵𝐶)
32a1i 11 . . 3 (𝑥𝐴 → (𝐶𝐵) = (𝐵𝐶))
43iuneq2i 4932 . 2 𝑥𝐴 (𝐶𝐵) = 𝑥𝐴 (𝐵𝐶)
5 incom 4177 . 2 ( 𝑥𝐴 𝐶𝐵) = (𝐵 𝑥𝐴 𝐶)
61, 4, 53eqtr4i 2854 1 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  cin 3934   ciun 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-in 3942  df-ss 3951  df-iun 4914
This theorem is referenced by:  2iunin  4990  resiun1  5867  tgrest  21697  metnrmlem3  23398  limciun  24421  uniin1  30231  disjunsn  30273  measinblem  31379  sstotbnd2  34935  subsaliuncl  42522  sge0iunmptlemre  42578
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