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Theorem iunin2 5079
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 5066 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
Assertion
Ref Expression
iunin2 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem iunin2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.42v 3181 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 𝑦𝐶))
2 elin 3963 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
32rexbii 3084 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
4 eliun 5005 . . . . 5 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
54anbi2i 621 . . . 4 ((𝑦𝐵𝑦 𝑥𝐴 𝐶) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 𝑦𝐶))
61, 3, 53bitr4i 302 . . 3 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦 𝑥𝐴 𝐶))
7 eliun 5005 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
8 elin 3963 . . 3 (𝑦 ∈ (𝐵 𝑥𝐴 𝐶) ↔ (𝑦𝐵𝑦 𝑥𝐴 𝐶))
96, 7, 83bitr4i 302 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ (𝐵 𝑥𝐴 𝐶))
109eqriv 2723 1 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1534  wcel 2099  wrex 3060  cin 3946   ciun 5001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rex 3061  df-v 3464  df-in 3954  df-iun 5003
This theorem is referenced by:  iunin1  5080  2iunin  5084  resiun2  6010  infssuni  9388  kmlem11  10203  cmpsublem  23394  cmpsub  23395  kgentopon  23533  metnrmlem3  24868  ovoliunlem1  25522  voliunlem1  25570  voliunlem2  25571  uniioombllem2  25603  uniioombllem4  25606  volsup2  25625  itg1addlem5  25721  itg1climres  25735  uniin2  32473  carsgclctunlem2  34153  cvmscld  35101  cnambfre  37369  ftc1anclem6  37399  heiborlem3  37514  carageniuncllem2  46143
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