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Theorem 2iunin 3879
Description: Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)
Assertion
Ref Expression
2iunin 𝑥𝐴 𝑦𝐵 (𝐶𝐷) = ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
Distinct variable groups:   𝑥,𝐵   𝑦,𝐶   𝑥,𝐷   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦)   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem 2iunin
StepHypRef Expression
1 iunin2 3876 . . . 4 𝑦𝐵 (𝐶𝐷) = (𝐶 𝑦𝐵 𝐷)
21a1i 9 . . 3 (𝑥𝐴 𝑦𝐵 (𝐶𝐷) = (𝐶 𝑦𝐵 𝐷))
32iuneq2i 3831 . 2 𝑥𝐴 𝑦𝐵 (𝐶𝐷) = 𝑥𝐴 (𝐶 𝑦𝐵 𝐷)
4 iunin1 3877 . 2 𝑥𝐴 (𝐶 𝑦𝐵 𝐷) = ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
53, 4eqtri 2160 1 𝑥𝐴 𝑦𝐵 (𝐶𝐷) = ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
Colors of variables: wff set class
Syntax hints:   = wceq 1331  wcel 1480  cin 3070   ciun 3813
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-iun 3815
This theorem is referenced by: (None)
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