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Theorem dfin2 3729
Description: An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 3728. Another version is given by dfin4 3733. (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 3111 . . . . . 6  |-  x  e. 
_V
2 eldif 3481 . . . . . 6  |-  ( x  e.  ( _V  \  B )  <->  ( x  e.  _V  /\  -.  x  e.  B ) )
31, 2mpbiran 911 . . . . 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 3481 . . 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 3687 1  |-  ( A  i^i  B )  =  ( A  \  ( _V  \  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3108    \ cdif 3468    i^i cin 3470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-v 3110  df-dif 3474  df-in 3478
This theorem is referenced by:  dfun3  3731  dfin3  3732  invdif  3734  difundi  3745  difindi  3747  difdif2  3750
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