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Theorem iunun 4756
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 3231 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (∃𝑥𝐴 𝑦𝐵 ∨ ∃𝑥𝐴 𝑦𝐶))
2 elun 3896 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
32rexbii 3179 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
4 eliun 4676 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
5 eliun 4676 . . . . 5 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
64, 5orbi12i 544 . . . 4 ((𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐶) ↔ (∃𝑥𝐴 𝑦𝐵 ∨ ∃𝑥𝐴 𝑦𝐶))
71, 3, 63bitr4i 292 . . 3 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐶))
8 eliun 4676 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
9 elun 3896 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶) ↔ (𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐶))
107, 8, 93bitr4i 292 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶))
1110eqriv 2757 1 𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wo 382   = wceq 1632  wcel 2139  wrex 3051  cun 3713   ciun 4672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-un 3720  df-iun 4674
This theorem is referenced by:  iununi  4762  oarec  7811  comppfsc  21537  uniiccdif  23546  bnj1415  31413  dftrpred4g  32039
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