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Theorem iundif2 4101
Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4088 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
Assertion
Ref Expression
iundif2  |-  U_ x  e.  A  ( B  \  C )  =  ( B  \  |^|_ x  e.  A  C )
Distinct variable group:    x, B
Allowed substitution hints:    A( x)    C( x)

Proof of Theorem iundif2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eldif 3275 . . . . 5  |-  ( y  e.  ( B  \  C )  <->  ( y  e.  B  /\  -.  y  e.  C ) )
21rexbii 2676 . . . 4  |-  ( E. x  e.  A  y  e.  ( B  \  C )  <->  E. x  e.  A  ( y  e.  B  /\  -.  y  e.  C ) )
3 r19.42v 2807 . . . 4  |-  ( E. x  e.  A  ( y  e.  B  /\  -.  y  e.  C
)  <->  ( y  e.  B  /\  E. x  e.  A  -.  y  e.  C ) )
4 rexnal 2662 . . . . . 6  |-  ( E. x  e.  A  -.  y  e.  C  <->  -.  A. x  e.  A  y  e.  C )
5 vex 2904 . . . . . . 7  |-  y  e. 
_V
6 eliin 4042 . . . . . . 7  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  C  <->  A. x  e.  A  y  e.  C ) )
75, 6ax-mp 8 . . . . . 6  |-  ( y  e.  |^|_ x  e.  A  C 
<-> 
A. x  e.  A  y  e.  C )
84, 7xchbinxr 303 . . . . 5  |-  ( E. x  e.  A  -.  y  e.  C  <->  -.  y  e.  |^|_ x  e.  A  C )
98anbi2i 676 . . . 4  |-  ( ( y  e.  B  /\  E. x  e.  A  -.  y  e.  C )  <->  ( y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C )
)
102, 3, 93bitri 263 . . 3  |-  ( E. x  e.  A  y  e.  ( B  \  C )  <->  ( y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C ) )
11 eliun 4041 . . 3  |-  ( y  e.  U_ x  e.  A  ( B  \  C )  <->  E. x  e.  A  y  e.  ( B  \  C ) )
12 eldif 3275 . . 3  |-  ( y  e.  ( B  \  |^|_ x  e.  A  C
)  <->  ( y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C ) )
1310, 11, 123bitr4i 269 . 2  |-  ( y  e.  U_ x  e.  A  ( B  \  C )  <->  y  e.  ( B  \  |^|_ x  e.  A  C )
)
1413eqriv 2386 1  |-  U_ x  e.  A  ( B  \  C )  =  ( B  \  |^|_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   E.wrex 2652   _Vcvv 2901    \ cdif 3262   U_ciun 4037   |^|_ciin 4038
This theorem is referenced by:  iuncld  17034  pnrmopn  17331  alexsublem  17998  bcth3  19155  iundifdifd  23858  iundifdif  23859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-rex 2657  df-v 2903  df-dif 3268  df-iun 4039  df-iin 4040
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