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Theorem iunin1 3845
 Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3834 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 3844 . 2 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
2 incom 3236 . . . 4 (𝐶𝐵) = (𝐵𝐶)
32a1i 9 . . 3 (𝑥𝐴 → (𝐶𝐵) = (𝐵𝐶))
43iuneq2i 3799 . 2 𝑥𝐴 (𝐶𝐵) = 𝑥𝐴 (𝐵𝐶)
5 incom 3236 . 2 ( 𝑥𝐴 𝐶𝐵) = (𝐵 𝑥𝐴 𝐶)
61, 4, 53eqtr4i 2146 1 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵)
 Colors of variables: wff set class Syntax hints:   = wceq 1314   ∈ wcel 1463   ∩ cin 3038  ∪ ciun 3781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-in 3045  df-ss 3052  df-iun 3783 This theorem is referenced by:  2iunin  3847  tgrest  12244
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