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Theorem 2iunin 3796
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 3793 . . . 4 𝑦𝐵 (𝐶𝐷) = (𝐶 𝑦𝐵 𝐷)
21a1i 9 . . 3 (𝑥𝐴 𝑦𝐵 (𝐶𝐷) = (𝐶 𝑦𝐵 𝐷))
32iuneq2i 3748 . 2 𝑥𝐴 𝑦𝐵 (𝐶𝐷) = 𝑥𝐴 (𝐶 𝑦𝐵 𝐷)
4 iunin1 3794 . 2 𝑥𝐴 (𝐶 𝑦𝐵 𝐷) = ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
53, 4eqtri 2108 1 𝑥𝐴 𝑦𝐵 (𝐶𝐷) = ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
Colors of variables: wff set class
Syntax hints:   = wceq 1289  wcel 1438  cin 2998   ciun 3730
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-iun 3732
This theorem is referenced by: (None)
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