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Theorem 2iunin 5099
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 5094 . . . 4 𝑦𝐵 (𝐶𝐷) = (𝐶 𝑦𝐵 𝐷)
21a1i 11 . . 3 (𝑥𝐴 𝑦𝐵 (𝐶𝐷) = (𝐶 𝑦𝐵 𝐷))
32iuneq2i 5036 . 2 𝑥𝐴 𝑦𝐵 (𝐶𝐷) = 𝑥𝐴 (𝐶 𝑦𝐵 𝐷)
4 iunin1 5095 . 2 𝑥𝐴 (𝐶 𝑦𝐵 𝐷) = ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
53, 4eqtri 2768 1 𝑥𝐴 𝑦𝐵 (𝐶𝐷) = ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2108  cin 3975   ciun 5015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-in 3983  df-ss 3993  df-iun 5017
This theorem is referenced by:  fpar  8157
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