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Theorem 2iunin 4159
Description: Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)
Assertion
Ref Expression
2iunin  |-  U_ x  e.  A  U_ y  e.  B  ( C  i^i  D )  =  ( U_ x  e.  A  C  i^i  U_ y  e.  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 4155 . . . 4  |-  U_ y  e.  B  ( C  i^i  D )  =  ( C  i^i  U_ y  e.  B  D )
21a1i 11 . . 3  |-  ( x  e.  A  ->  U_ y  e.  B  ( C  i^i  D )  =  ( C  i^i  U_ y  e.  B  D )
)
32iuneq2i 4111 . 2  |-  U_ x  e.  A  U_ y  e.  B  ( C  i^i  D )  =  U_ x  e.  A  ( C  i^i  U_ y  e.  B  D )
4 iunin1 4156 . 2  |-  U_ x  e.  A  ( C  i^i  U_ y  e.  B  D )  =  (
U_ x  e.  A  C  i^i  U_ y  e.  B  D )
53, 4eqtri 2456 1  |-  U_ x  e.  A  U_ y  e.  B  ( C  i^i  D )  =  ( U_ x  e.  A  C  i^i  U_ y  e.  B  D )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725    i^i cin 3319   U_ciun 4093
This theorem is referenced by:  fpar  6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958  df-in 3327  df-ss 3334  df-iun 4095
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