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Theorem iinun2 4962
 Description: Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4950 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
Assertion
Ref Expression
iinun2 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem iinun2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.32v 3295 . . . 4 (∀𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (𝑦𝐵 ∨ ∀𝑥𝐴 𝑦𝐶))
2 elun 4079 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
32ralbii 3133 . . . 4 (∀𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∀𝑥𝐴 (𝑦𝐵𝑦𝐶))
4 eliin 4890 . . . . . 6 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶))
54elv 3447 . . . . 5 (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶)
65orbi2i 910 . . . 4 ((𝑦𝐵𝑦 𝑥𝐴 𝐶) ↔ (𝑦𝐵 ∨ ∀𝑥𝐴 𝑦𝐶))
71, 3, 63bitr4i 306 . . 3 (∀𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦 𝑥𝐴 𝐶))
8 eliin 4890 . . . 4 (𝑦 ∈ V → (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶)))
98elv 3447 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∀𝑥𝐴 𝑦 ∈ (𝐵𝐶))
10 elun 4079 . . 3 (𝑦 ∈ (𝐵 𝑥𝐴 𝐶) ↔ (𝑦𝐵𝑦 𝑥𝐴 𝐶))
117, 9, 103bitr4i 306 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ (𝐵 𝑥𝐴 𝐶))
1211eqriv 2795 1 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3442   ∪ cun 3881  ∩ ciin 4886 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3444  df-un 3888  df-iin 4888 This theorem is referenced by: (None)
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