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Theorem iindif2 4350
Description: Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4334 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 3886 . . . 4  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  (
y  e.  B  /\  -.  y  e.  C
)  <->  ( y  e.  B  /\  A. x  e.  A  -.  y  e.  C ) ) )
2 eldif 3449 . . . . . 6  |-  ( y  e.  ( B  \  C )  <->  ( y  e.  B  /\  -.  y  e.  C ) )
32bicomi 202 . . . . 5  |-  ( ( y  e.  B  /\  -.  y  e.  C
)  <->  y  e.  ( B  \  C ) )
43ralbii 2839 . . . 4  |-  ( A. x  e.  A  (
y  e.  B  /\  -.  y  e.  C
)  <->  A. x  e.  A  y  e.  ( B  \  C ) )
5 ralnex 2852 . . . . . 6  |-  ( A. x  e.  A  -.  y  e.  C  <->  -.  E. x  e.  A  y  e.  C )
6 eliun 4286 . . . . . 6  |-  ( y  e.  U_ x  e.  A  C  <->  E. x  e.  A  y  e.  C )
75, 6xchbinxr 311 . . . . 5  |-  ( A. x  e.  A  -.  y  e.  C  <->  -.  y  e.  U_ x  e.  A  C )
87anbi2i 694 . . . 4  |-  ( ( y  e.  B  /\  A. x  e.  A  -.  y  e.  C )  <->  ( y  e.  B  /\  -.  y  e.  U_ x  e.  A  C )
)
91, 4, 83bitr3g 287 . . 3  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  y  e.  ( B  \  C
)  <->  ( y  e.  B  /\  -.  y  e.  U_ x  e.  A  C ) ) )
10 vex 3081 . . . 4  |-  y  e. 
_V
11 eliin 4287 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  ( B  \  C )  <->  A. x  e.  A  y  e.  ( B  \  C ) ) )
1210, 11ax-mp 5 . . 3  |-  ( y  e.  |^|_ x  e.  A  ( B  \  C )  <->  A. x  e.  A  y  e.  ( B  \  C ) )
13 eldif 3449 . . 3  |-  ( y  e.  ( B  \  U_ x  e.  A  C )  <->  ( y  e.  B  /\  -.  y  e.  U_ x  e.  A  C ) )
149, 12, 133bitr4g 288 . 2  |-  ( A  =/=  (/)  ->  ( y  e.  |^|_ x  e.  A  ( B  \  C )  <-> 
y  e.  ( B 
\  U_ x  e.  A  C ) ) )
1514eqrdv 2451 1  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( B  \  C )  =  ( B  \  U_ x  e.  A  C )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800   _Vcvv 3078    \ cdif 3436   (/)c0 3748   U_ciun 4282   |^|_ciin 4283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-v 3080  df-dif 3442  df-nul 3749  df-iun 4284  df-iin 4285
This theorem is referenced by:  iinvdif  4353  iincld  18778  clsval2  18789  mretopd  18831  hauscmplem  19144  cmpfi  19146
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