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Theorem iindif2 4152
Description: Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4136 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)
Assertion
Ref Expression
iindif2  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( B  \  C )  =  ( B  \  U_ x  e.  A  C )
)
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem iindif2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 r19.28zv 3715 . . . 4  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  (
y  e.  B  /\  -.  y  e.  C
)  <->  ( y  e.  B  /\  A. x  e.  A  -.  y  e.  C ) ) )
2 eldif 3322 . . . . . 6  |-  ( y  e.  ( B  \  C )  <->  ( y  e.  B  /\  -.  y  e.  C ) )
32bicomi 194 . . . . 5  |-  ( ( y  e.  B  /\  -.  y  e.  C
)  <->  y  e.  ( B  \  C ) )
43ralbii 2721 . . . 4  |-  ( A. x  e.  A  (
y  e.  B  /\  -.  y  e.  C
)  <->  A. x  e.  A  y  e.  ( B  \  C ) )
5 ralnex 2707 . . . . . 6  |-  ( A. x  e.  A  -.  y  e.  C  <->  -.  E. x  e.  A  y  e.  C )
6 eliun 4089 . . . . . 6  |-  ( y  e.  U_ x  e.  A  C  <->  E. x  e.  A  y  e.  C )
75, 6xchbinxr 303 . . . . 5  |-  ( A. x  e.  A  -.  y  e.  C  <->  -.  y  e.  U_ x  e.  A  C )
87anbi2i 676 . . . 4  |-  ( ( y  e.  B  /\  A. x  e.  A  -.  y  e.  C )  <->  ( y  e.  B  /\  -.  y  e.  U_ x  e.  A  C )
)
91, 4, 83bitr3g 279 . . 3  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  y  e.  ( B  \  C
)  <->  ( y  e.  B  /\  -.  y  e.  U_ x  e.  A  C ) ) )
10 vex 2951 . . . 4  |-  y  e. 
_V
11 eliin 4090 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  ( B  \  C )  <->  A. x  e.  A  y  e.  ( B  \  C ) ) )
1210, 11ax-mp 8 . . 3  |-  ( y  e.  |^|_ x  e.  A  ( B  \  C )  <->  A. x  e.  A  y  e.  ( B  \  C ) )
13 eldif 3322 . . 3  |-  ( y  e.  ( B  \  U_ x  e.  A  C )  <->  ( y  e.  B  /\  -.  y  e.  U_ x  e.  A  C ) )
149, 12, 133bitr4g 280 . 2  |-  ( A  =/=  (/)  ->  ( y  e.  |^|_ x  e.  A  ( B  \  C )  <-> 
y  e.  ( B 
\  U_ x  e.  A  C ) ) )
1514eqrdv 2433 1  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( B  \  C )  =  ( B  \  U_ x  e.  A  C )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948    \ cdif 3309   (/)c0 3620   U_ciun 4085   |^|_ciin 4086
This theorem is referenced by:  iincld  17095  clsval2  17106  mretopd  17148  hauscmplem  17461  cmpfi  17463
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-nul 3621  df-iun 4087  df-iin 4088
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