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Theorem 2iunin 3970
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 3966 . . . 4  |-  U_ y  e.  B  ( C  i^i  D )  =  ( C  i^i  U_ y  e.  B  D )
21a1i 10 . . 3  |-  ( x  e.  A  ->  U_ y  e.  B  ( C  i^i  D )  =  ( C  i^i  U_ y  e.  B  D )
)
32iuneq2i 3923 . 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 3967 . 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 2303 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 1623    e. wcel 1684    i^i cin 3151   U_ciun 3905
This theorem is referenced by:  fpar  6222
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-iun 3907
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