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Theorem iunun 5097
Description: Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
iunun 𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶)

Proof of Theorem iunun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.43 3119 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵 ∨ ∃𝑥𝐴 𝑦𝐶))
2 elun 4162 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
32rexbii 3091 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
4 eliun 4999 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
5 eliun 4999 . . . . 5 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
64, 5orbi12i 914 . . . 4 ((𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐶) ↔ (∃𝑥𝐴 𝑦𝐵 ∨ ∃𝑥𝐴 𝑦𝐶))
71, 3, 63bitr4i 303 . . 3 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐶))
8 eliun 4999 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
9 elun 4162 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶) ↔ (𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐶))
107, 8, 93bitr4i 303 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶))
1110eqriv 2731 1 𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wo 847   = wceq 1536  wcel 2105  wrex 3067  cun 3960   ciun 4995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-v 3479  df-un 3967  df-iun 4997
This theorem is referenced by:  iununi  5103  oarec  8598  comppfsc  23555  uniiccdif  25626  bnj1415  35030
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