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Theorem iindif2 4347
Description: Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4331 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 3864 . . . 4  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  (
y  e.  B  /\  -.  y  e.  C
)  <->  ( y  e.  B  /\  A. x  e.  A  -.  y  e.  C ) ) )
2 eldif 3414 . . . . . 6  |-  ( y  e.  ( B  \  C )  <->  ( y  e.  B  /\  -.  y  e.  C ) )
32bicomi 206 . . . . 5  |-  ( ( y  e.  B  /\  -.  y  e.  C
)  <->  y  e.  ( B  \  C ) )
43ralbii 2819 . . . 4  |-  ( A. x  e.  A  (
y  e.  B  /\  -.  y  e.  C
)  <->  A. x  e.  A  y  e.  ( B  \  C ) )
5 ralnex 2834 . . . . . 6  |-  ( A. x  e.  A  -.  y  e.  C  <->  -.  E. x  e.  A  y  e.  C )
6 eliun 4283 . . . . . 6  |-  ( y  e.  U_ x  e.  A  C  <->  E. x  e.  A  y  e.  C )
75, 6xchbinxr 313 . . . . 5  |-  ( A. x  e.  A  -.  y  e.  C  <->  -.  y  e.  U_ x  e.  A  C )
87anbi2i 700 . . . 4  |-  ( ( y  e.  B  /\  A. x  e.  A  -.  y  e.  C )  <->  ( y  e.  B  /\  -.  y  e.  U_ x  e.  A  C )
)
91, 4, 83bitr3g 291 . . 3  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  y  e.  ( B  \  C
)  <->  ( y  e.  B  /\  -.  y  e.  U_ x  e.  A  C ) ) )
10 vex 3048 . . . 4  |-  y  e. 
_V
11 eliin 4284 . . . 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 3414 . . 3  |-  ( y  e.  ( B  \  U_ x  e.  A  C )  <->  ( y  e.  B  /\  -.  y  e.  U_ x  e.  A  C ) )
149, 12, 133bitr4g 292 . 2  |-  ( A  =/=  (/)  ->  ( y  e.  |^|_ x  e.  A  ( B  \  C )  <-> 
y  e.  ( B 
\  U_ x  e.  A  C ) ) )
1514eqrdv 2449 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 188    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738   _Vcvv 3045    \ cdif 3401   (/)c0 3731   U_ciun 4278   |^|_ciin 4279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-v 3047  df-dif 3407  df-nul 3732  df-iun 4280  df-iin 4281
This theorem is referenced by:  iinvdif  4350  iincld  20054  clsval2  20065  mretopd  20108  hauscmplem  20421  cmpfi  20423  sigapildsyslem  28983  saliincl  38186
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