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Theorem iunin1 3721
 Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3710 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
iunin1 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem iunin1
StepHypRef Expression
1 iunin2 3720 . 2 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
2 incom 3129 . . . 4 (𝐶𝐵) = (𝐵𝐶)
32a1i 9 . . 3 (𝑥𝐴 → (𝐶𝐵) = (𝐵𝐶))
43iuneq2i 3675 . 2 𝑥𝐴 (𝐶𝐵) = 𝑥𝐴 (𝐵𝐶)
5 incom 3129 . 2 ( 𝑥𝐴 𝐶𝐵) = (𝐵 𝑥𝐴 𝐶)
61, 4, 53eqtr4i 2070 1 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵)
 Colors of variables: wff set class Syntax hints:   = wceq 1243   ∈ wcel 1393   ∩ cin 2916  ∪ ciun 3657 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-iun 3659 This theorem is referenced by:  2iunin  3723
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