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Theorem dfin2 3569
Description: An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 3568. Another version is given by dfin4 3573. (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 2951 . . . . . 6  |-  x  e. 
_V
2 eldif 3322 . . . . . 6  |-  ( x  e.  ( _V  \  B )  <->  ( x  e.  _V  /\  -.  x  e.  B ) )
31, 2mpbiran 885 . . . . 5  |-  ( x  e.  ( _V  \  B )  <->  -.  x  e.  B )
43con2bii 323 . . . 4  |-  ( x  e.  B  <->  -.  x  e.  ( _V  \  B
) )
54anbi2i 676 . . 3  |-  ( ( x  e.  A  /\  x  e.  B )  <->  ( x  e.  A  /\  -.  x  e.  ( _V  \  B ) ) )
6 eldif 3322 . . 3  |-  ( x  e.  ( A  \ 
( _V  \  B
) )  <->  ( x  e.  A  /\  -.  x  e.  ( _V  \  B
) ) )
75, 6bitr4i 244 . 2  |-  ( ( x  e.  A  /\  x  e.  B )  <->  x  e.  ( A  \ 
( _V  \  B
) ) )
87ineqri 3526 1  |-  ( A  i^i  B )  =  ( A  \  ( _V  \  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309    i^i cin 3311
This theorem is referenced by:  dfun3  3571  dfin3  3572  invdif  3574  difundi  3585  difindi  3587  difdif2  3590
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-dif 3315  df-in 3319
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