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Theorem iunin2 3846
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3836 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
Assertion
Ref Expression
iunin2  |-  U_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  U_ x  e.  A  C )
Distinct variable group:    x, B
Allowed substitution hints:    A( x)    C( x)

Proof of Theorem iunin2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 r19.42v 2565 . . . 4  |-  ( E. x  e.  A  ( y  e.  B  /\  y  e.  C )  <->  ( y  e.  B  /\  E. x  e.  A  y  e.  C ) )
2 elin 3229 . . . . 5  |-  ( y  e.  ( B  i^i  C )  <->  ( y  e.  B  /\  y  e.  C ) )
32rexbii 2419 . . . 4  |-  ( E. x  e.  A  y  e.  ( B  i^i  C )  <->  E. x  e.  A  ( y  e.  B  /\  y  e.  C
) )
4 eliun 3787 . . . . 5  |-  ( y  e.  U_ x  e.  A  C  <->  E. x  e.  A  y  e.  C )
54anbi2i 452 . . . 4  |-  ( ( y  e.  B  /\  y  e.  U_ x  e.  A  C )  <->  ( y  e.  B  /\  E. x  e.  A  y  e.  C ) )
61, 3, 53bitr4i 211 . . 3  |-  ( E. x  e.  A  y  e.  ( B  i^i  C )  <->  ( y  e.  B  /\  y  e. 
U_ x  e.  A  C ) )
7 eliun 3787 . . 3  |-  ( y  e.  U_ x  e.  A  ( B  i^i  C )  <->  E. x  e.  A  y  e.  ( B  i^i  C ) )
8 elin 3229 . . 3  |-  ( y  e.  ( B  i^i  U_ x  e.  A  C
)  <->  ( y  e.  B  /\  y  e. 
U_ x  e.  A  C ) )
96, 7, 83bitr4i 211 . 2  |-  ( y  e.  U_ x  e.  A  ( B  i^i  C )  <->  y  e.  ( B  i^i  U_ x  e.  A  C )
)
109eqriv 2114 1  |-  U_ x  e.  A  ( B  i^i  C )  =  ( B  i^i  U_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1316    e. wcel 1465   E.wrex 2394    i^i cin 3040   U_ciun 3783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-in 3047  df-iun 3785
This theorem is referenced by:  iunin1  3847  2iunin  3849  resiun1  4808  resiun2  4809
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