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Theorem iunun 3861
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 2566 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵 ∨ ∃𝑥𝐴 𝑦𝐶))
2 elun 3187 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
32rexbii 2419 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
4 eliun 3787 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
5 eliun 3787 . . . . 5 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
64, 5orbi12i 738 . . . 4 ((𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐶) ↔ (∃𝑥𝐴 𝑦𝐵 ∨ ∃𝑥𝐴 𝑦𝐶))
71, 3, 63bitr4i 211 . . 3 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐶))
8 eliun 3787 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
9 elun 3187 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶) ↔ (𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐶))
107, 8, 93bitr4i 211 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶))
1110eqriv 2114 1 𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wo 682   = wceq 1316  wcel 1465  wrex 2394  cun 3039   ciun 3783
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-iun 3785
This theorem is referenced by: (None)
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