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Theorem 2iunin 4620
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 4616 . . . 4 𝑦𝐵 (𝐶𝐷) = (𝐶 𝑦𝐵 𝐷)
21a1i 11 . . 3 (𝑥𝐴 𝑦𝐵 (𝐶𝐷) = (𝐶 𝑦𝐵 𝐷))
32iuneq2i 4571 . 2 𝑥𝐴 𝑦𝐵 (𝐶𝐷) = 𝑥𝐴 (𝐶 𝑦𝐵 𝐷)
4 iunin1 4617 . 2 𝑥𝐴 (𝐶 𝑦𝐵 𝐷) = ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
53, 4eqtri 2673 1 𝑥𝐴 𝑦𝐵 (𝐶𝐷) = ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  cin 3606   ciun 4552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-in 3614  df-ss 3621  df-iun 4554
This theorem is referenced by:  fpar  7326
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