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Theorem iundif2 3970
Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 3957 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
Dummy variable  y is distinct from all other variables.
Allowed substitution hints:    A( x)    C( x)

Proof of Theorem iundif2
StepHypRef Expression
1 eldif 3163 . . . . 5  |-  ( y  e.  ( B  \  C )  <->  ( y  e.  B  /\  -.  y  e.  C ) )
21rexbii 2569 . . . 4  |-  ( E. x  e.  A  y  e.  ( B  \  C )  <->  E. x  e.  A  ( y  e.  B  /\  -.  y  e.  C ) )
3 r19.42v 2695 . . . 4  |-  ( E. x  e.  A  ( y  e.  B  /\  -.  y  e.  C
)  <->  ( y  e.  B  /\  E. x  e.  A  -.  y  e.  C ) )
4 rexnal 2555 . . . . . 6  |-  ( E. x  e.  A  -.  y  e.  C  <->  -.  A. x  e.  A  y  e.  C )
5 vex 2792 . . . . . . 7  |-  y  e. 
_V
6 eliin 3911 . . . . . . 7  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  C  <->  A. x  e.  A  y  e.  C ) )
75, 6ax-mp 10 . . . . . 6  |-  ( y  e.  |^|_ x  e.  A  C 
<-> 
A. x  e.  A  y  e.  C )
84, 7xchbinxr 304 . . . . 5  |-  ( E. x  e.  A  -.  y  e.  C  <->  -.  y  e.  |^|_ x  e.  A  C )
98anbi2i 677 . . . 4  |-  ( ( y  e.  B  /\  E. x  e.  A  -.  y  e.  C )  <->  ( y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C )
)
102, 3, 93bitri 264 . . 3  |-  ( E. x  e.  A  y  e.  ( B  \  C )  <->  ( y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C ) )
11 eliun 3910 . . 3  |-  ( y  e.  U_ x  e.  A  ( B  \  C )  <->  E. x  e.  A  y  e.  ( B  \  C ) )
12 eldif 3163 . . 3  |-  ( y  e.  ( B  \  |^|_ x  e.  A  C
)  <->  ( y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C ) )
1310, 11, 123bitr4i 270 . 2  |-  ( y  e.  U_ x  e.  A  ( B  \  C )  <->  y  e.  ( B  \  |^|_ x  e.  A  C )
)
1413eqriv 2281 1  |-  U_ x  e.  A  ( B  \  C )  =  ( B  \  |^|_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685   A.wral 2544   E.wrex 2545   _Vcvv 2789    \ cdif 3150   U_ciun 3906   |^|_ciin 3907
This theorem is referenced by:  iuncld  16776  pnrmopn  17065  alexsublem  17732  bcth3  18747
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ral 2549  df-rex 2550  df-v 2791  df-dif 3156  df-iun 3908  df-iin 3909
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