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Theorem iunxun 3762
Description: Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
iunxun 𝑥 ∈ (𝐴𝐵)𝐶 = ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶)

Proof of Theorem iunxun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 rexun 3150 . . . 4 (∃𝑥 ∈ (𝐴𝐵)𝑦𝐶 ↔ (∃𝑥𝐴 𝑦𝐶 ∨ ∃𝑥𝐵 𝑦𝐶))
2 eliun 3688 . . . . 5 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
3 eliun 3688 . . . . 5 (𝑦 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝑦𝐶)
42, 3orbi12i 691 . . . 4 ((𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶) ↔ (∃𝑥𝐴 𝑦𝐶 ∨ ∃𝑥𝐵 𝑦𝐶))
51, 4bitr4i 180 . . 3 (∃𝑥 ∈ (𝐴𝐵)𝑦𝐶 ↔ (𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶))
6 eliun 3688 . . 3 (𝑦 𝑥 ∈ (𝐴𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴𝐵)𝑦𝐶)
7 elun 3111 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶) ↔ (𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶))
85, 6, 73bitr4i 205 . 2 (𝑦 𝑥 ∈ (𝐴𝐵)𝐶𝑦 ∈ ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶))
98eqriv 2053 1 𝑥 ∈ (𝐴𝐵)𝐶 = ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
Colors of variables: wff set class
Syntax hints:  wo 639   = wceq 1259  wcel 1409  wrex 2324  cun 2942   ciun 3684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-iun 3686
This theorem is referenced by:  iunsuc  4184  rdgisuc1  6001  oasuc  6074  omsuc  6081
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