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Theorem dfin2 3686
Description: An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 3685. Another version is given by dfin4 3690. (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 3073 . . . . . 6  |-  x  e. 
_V
2 eldif 3438 . . . . . 6  |-  ( x  e.  ( _V  \  B )  <->  ( x  e.  _V  /\  -.  x  e.  B ) )
31, 2mpbiran 909 . . . . 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 3438 . . 3  |-  ( x  e.  ( A  \ 
( _V  \  B
) )  <->  ( x  e.  A  /\  -.  x  e.  ( _V  \  B
) ) )
75, 6bitr4i 252 . 2  |-  ( ( x  e.  A  /\  x  e.  B )  <->  x  e.  ( A  \ 
( _V  \  B
) ) )
87ineqri 3644 1  |-  ( A  i^i  B )  =  ( A  \  ( _V  \  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3070    \ cdif 3425    i^i cin 3427
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 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3072  df-dif 3431  df-in 3435
This theorem is referenced by:  dfun3  3688  dfin3  3689  invdif  3691  difundi  3702  difindi  3704  difdif2  3707
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