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Theorem 2iunin 4034
 Description: Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)
Assertion
Ref Expression
2iunin x A y B (CD) = (x A Cy B 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 4030 . . . 4 y B (CD) = (Cy B D)
21a1i 10 . . 3 (x Ay B (CD) = (Cy B D))
32iuneq2i 3987 . 2 x A y B (CD) = x A (Cy B D)
4 iunin1 4031 . 2 x A (Cy B D) = (x A Cy B D)
53, 4eqtri 2373 1 x A y B (CD) = (x A Cy B D)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∈ wcel 1710   ∩ cin 3208  ∪ciun 3969 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-iun 3971 This theorem is referenced by: (None)
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