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Theorem iunin2 3748
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3738 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
Assertion
Ref Expression
iunin2 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem iunin2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.42v 2484 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 𝑦𝐶))
2 elin 3154 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
32rexbii 2348 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
4 eliun 3689 . . . . 5 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
54anbi2i 438 . . . 4 ((𝑦𝐵𝑦 𝑥𝐴 𝐶) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 𝑦𝐶))
61, 3, 53bitr4i 205 . . 3 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦 𝑥𝐴 𝐶))
7 eliun 3689 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
8 elin 3154 . . 3 (𝑦 ∈ (𝐵 𝑥𝐴 𝐶) ↔ (𝑦𝐵𝑦 𝑥𝐴 𝐶))
96, 7, 83bitr4i 205 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ (𝐵 𝑥𝐴 𝐶))
109eqriv 2053 1 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  wcel 1409  wrex 2324  cin 2944   ciun 3685
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-in 2952  df-iun 3687
This theorem is referenced by:  iunin1  3749  2iunin  3751  resiun1  4658  resiun2  4659
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