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Theorem iunun 3951
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 2628 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵 ∨ ∃𝑥𝐴 𝑦𝐶))
2 elun 3268 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
32rexbii 2477 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
4 eliun 3877 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
5 eliun 3877 . . . . 5 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
64, 5orbi12i 759 . . . 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 3877 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
9 elun 3268 . . 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 2167 1 𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wo 703   = wceq 1348  wcel 2141  wrex 2449  cun 3119   ciun 3873
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-iun 3875
This theorem is referenced by: (None)
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