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Theorem 2iunin 4964
 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 4959 . . . 4 𝑦𝐵 (𝐶𝐷) = (𝐶 𝑦𝐵 𝐷)
21a1i 11 . . 3 (𝑥𝐴 𝑦𝐵 (𝐶𝐷) = (𝐶 𝑦𝐵 𝐷))
32iuneq2i 4905 . 2 𝑥𝐴 𝑦𝐵 (𝐶𝐷) = 𝑥𝐴 (𝐶 𝑦𝐵 𝐷)
4 iunin1 4960 . 2 𝑥𝐴 (𝐶 𝑦𝐵 𝐷) = ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
53, 4eqtri 2782 1 𝑥𝐴 𝑦𝐵 (𝐶𝐷) = ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1539   ∈ wcel 2112   ∩ cin 3858  ∪ ciun 4884 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-in 3866  df-ss 3876  df-iun 4886 This theorem is referenced by:  fpar  7817
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