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Theorem dfin2 3688
Description: An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 3687. Another version is given by dfin4 3692. (Contributed by NM, 10-Jun-2004.)
Assertion
Ref Expression
dfin2  |-  ( A  i^i  B )  =  ( A  \  ( _V  \  B ) )

Proof of Theorem dfin2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 3064 . . . . . 6  |-  x  e. 
_V
2 eldif 3426 . . . . . 6  |-  ( x  e.  ( _V  \  B )  <->  ( x  e.  _V  /\  -.  x  e.  B ) )
31, 2mpbiran 921 . . . . 5  |-  ( x  e.  ( _V  \  B )  <->  -.  x  e.  B )
43con2bii 332 . . . 4  |-  ( x  e.  B  <->  -.  x  e.  ( _V  \  B
) )
54anbi2i 694 . . 3  |-  ( ( x  e.  A  /\  x  e.  B )  <->  ( x  e.  A  /\  -.  x  e.  ( _V  \  B ) ) )
6 eldif 3426 . . 3  |-  ( x  e.  ( A  \ 
( _V  \  B
) )  <->  ( x  e.  A  /\  -.  x  e.  ( _V  \  B
) ) )
75, 6bitr4i 254 . 2  |-  ( ( x  e.  A  /\  x  e.  B )  <->  x  e.  ( A  \ 
( _V  \  B
) ) )
87ineqri 3635 1  |-  ( A  i^i  B )  =  ( A  \  ( _V  \  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1407    e. wcel 1844   _Vcvv 3061    \ cdif 3413    i^i cin 3415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-v 3063  df-dif 3419  df-in 3423
This theorem is referenced by:  dfun3  3690  dfin3  3691  invdif  3693  difundi  3704  difindi  3706  difdif2  3709
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