ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iindif2m Unicode version

Theorem iindif2m 3969
Description: Indexed intersection of class difference. Compare to Theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
iindif2m  |-  ( E. x  x  e.  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 iindif2m
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 r19.28mv 3530 . . . 4  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ( y  e.  B  /\  -.  y  e.  C )  <->  ( y  e.  B  /\  A. x  e.  A  -.  y  e.  C ) ) )
2 eldif 3153 . . . . . 6  |-  ( y  e.  ( B  \  C )  <->  ( y  e.  B  /\  -.  y  e.  C ) )
32bicomi 132 . . . . 5  |-  ( ( y  e.  B  /\  -.  y  e.  C
)  <->  y  e.  ( B  \  C ) )
43ralbii 2496 . . . 4  |-  ( A. x  e.  A  (
y  e.  B  /\  -.  y  e.  C
)  <->  A. x  e.  A  y  e.  ( B  \  C ) )
5 ralnex 2478 . . . . . 6  |-  ( A. x  e.  A  -.  y  e.  C  <->  -.  E. x  e.  A  y  e.  C )
6 eliun 3905 . . . . . 6  |-  ( y  e.  U_ x  e.  A  C  <->  E. x  e.  A  y  e.  C )
75, 6xchbinxr 684 . . . . 5  |-  ( A. x  e.  A  -.  y  e.  C  <->  -.  y  e.  U_ x  e.  A  C )
87anbi2i 457 . . . 4  |-  ( ( y  e.  B  /\  A. x  e.  A  -.  y  e.  C )  <->  ( y  e.  B  /\  -.  y  e.  U_ x  e.  A  C )
)
91, 4, 83bitr3g 222 . . 3  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  y  e.  ( B  \  C )  <-> 
( y  e.  B  /\  -.  y  e.  U_ x  e.  A  C
) ) )
10 vex 2755 . . . 4  |-  y  e. 
_V
11 eliin 3906 . . . 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 3153 . . 3  |-  ( y  e.  ( B  \  U_ x  e.  A  C )  <->  ( y  e.  B  /\  -.  y  e.  U_ x  e.  A  C ) )
149, 12, 133bitr4g 223 . 2  |-  ( E. x  x  e.  A  ->  ( y  e.  |^|_ x  e.  A  ( B 
\  C )  <->  y  e.  ( B  \  U_ x  e.  A  C )
) )
1514eqrdv 2187 1  |-  ( E. x  x  e.  A  -> 
|^|_ x  e.  A  ( B  \  C )  =  ( B  \  U_ x  e.  A  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364   E.wex 1503    e. wcel 2160   A.wral 2468   E.wrex 2469   _Vcvv 2752    \ cdif 3141   U_ciun 3901   |^|_ciin 3902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-iun 3903  df-iin 3904
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator