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Theorem iundif2 3969
Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 3956 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 3162 . . . . 5  |-  ( y  e.  ( B  \  C )  <->  ( y  e.  B  /\  -.  y  e.  C ) )
21rexbii 2568 . . . 4  |-  ( E. x  e.  A  y  e.  ( B  \  C )  <->  E. x  e.  A  ( y  e.  B  /\  -.  y  e.  C ) )
3 r19.42v 2694 . . . 4  |-  ( E. x  e.  A  ( y  e.  B  /\  -.  y  e.  C
)  <->  ( y  e.  B  /\  E. x  e.  A  -.  y  e.  C ) )
4 rexnal 2554 . . . . . 6  |-  ( E. x  e.  A  -.  y  e.  C  <->  -.  A. x  e.  A  y  e.  C )
5 vex 2791 . . . . . . 7  |-  y  e. 
_V
6 eliin 3910 . . . . . . 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 302 . . . . 5  |-  ( E. x  e.  A  -.  y  e.  C  <->  -.  y  e.  |^|_ x  e.  A  C )
98anbi2i 675 . . . 4  |-  ( ( y  e.  B  /\  E. x  e.  A  -.  y  e.  C )  <->  ( y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C )
)
102, 3, 93bitri 262 . . 3  |-  ( E. x  e.  A  y  e.  ( B  \  C )  <->  ( y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C ) )
11 eliun 3909 . . 3  |-  ( y  e.  U_ x  e.  A  ( B  \  C )  <->  E. x  e.  A  y  e.  ( B  \  C ) )
12 eldif 3162 . . 3  |-  ( y  e.  ( B  \  |^|_ x  e.  A  C
)  <->  ( y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C ) )
1310, 11, 123bitr4i 268 . 2  |-  ( y  e.  U_ x  e.  A  ( B  \  C )  <->  y  e.  ( B  \  |^|_ x  e.  A  C )
)
1413eqriv 2280 1  |-  U_ x  e.  A  ( B  \  C )  =  ( B  \  |^|_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    \ cdif 3149   U_ciun 3905   |^|_ciin 3906
This theorem is referenced by:  iuncld  16782  pnrmopn  17071  alexsublem  17738  bcth3  18753  iundifdifd  23159  iundifdif  23160
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-dif 3155  df-iun 3907  df-iin 3908
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