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Theorem iindif2 3912
Description: Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 3896 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
StepHypRef Expression
1 r19.28zv 3491 . . . 4  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  (
y  e.  B  /\  -.  y  e.  C
)  <->  ( y  e.  B  /\  A. x  e.  A  -.  y  e.  C ) ) )
2 eldif 3104 . . . . . 6  |-  ( y  e.  ( B  \  C )  <->  ( y  e.  B  /\  -.  y  e.  C ) )
32bicomi 195 . . . . 5  |-  ( ( y  e.  B  /\  -.  y  e.  C
)  <->  y  e.  ( B  \  C ) )
43ralbii 2538 . . . 4  |-  ( A. x  e.  A  (
y  e.  B  /\  -.  y  e.  C
)  <->  A. x  e.  A  y  e.  ( B  \  C ) )
5 ralnex 2524 . . . . . 6  |-  ( A. x  e.  A  -.  y  e.  C  <->  -.  E. x  e.  A  y  e.  C )
6 eliun 3850 . . . . . 6  |-  ( y  e.  U_ x  e.  A  C  <->  E. x  e.  A  y  e.  C )
75, 6xchbinxr 304 . . . . 5  |-  ( A. x  e.  A  -.  y  e.  C  <->  -.  y  e.  U_ x  e.  A  C )
87anbi2i 678 . . . 4  |-  ( ( y  e.  B  /\  A. x  e.  A  -.  y  e.  C )  <->  ( y  e.  B  /\  -.  y  e.  U_ x  e.  A  C )
)
91, 4, 83bitr3g 280 . . 3  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  y  e.  ( B  \  C
)  <->  ( y  e.  B  /\  -.  y  e.  U_ x  e.  A  C ) ) )
10 vex 2743 . . . 4  |-  y  e. 
_V
11 eliin 3851 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  ( B  \  C )  <->  A. x  e.  A  y  e.  ( B  \  C ) ) )
1210, 11ax-mp 10 . . 3  |-  ( y  e.  |^|_ x  e.  A  ( B  \  C )  <->  A. x  e.  A  y  e.  ( B  \  C ) )
13 eldif 3104 . . 3  |-  ( y  e.  ( B  \  U_ x  e.  A  C )  <->  ( y  e.  B  /\  -.  y  e.  U_ x  e.  A  C ) )
149, 12, 133bitr4g 281 . 2  |-  ( A  =/=  (/)  ->  ( y  e.  |^|_ x  e.  A  ( B  \  C )  <-> 
y  e.  ( B 
\  U_ x  e.  A  C ) ) )
1514eqrdv 2254 1  |-  ( A  =/=  (/)  ->  |^|_ x  e.  A  ( B  \  C )  =  ( B  \  U_ x  e.  A  C )
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   E.wrex 2517   _Vcvv 2740    \ cdif 3091   (/)c0 3397   U_ciun 3846   |^|_ciin 3847
This theorem is referenced by:  iincld  16703  clsval2  16714  mretopd  16756  hauscmplem  17060  cmpfi  17062
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-v 2742  df-dif 3097  df-nul 3398  df-iun 3848  df-iin 3849
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