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Theorem ddif 3597
Description: Double complement under universal class. Exercise 4.10(s) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)
Assertion
Ref Expression
ddif  |-  ( _V 
\  ( _V  \  A ) )  =  A

Proof of Theorem ddif
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 3084 . . . . 5  |-  x  e. 
_V
2 eldif 3446 . . . . 5  |-  ( x  e.  ( _V  \  A )  <->  ( x  e.  _V  /\  -.  x  e.  A ) )
31, 2mpbiran 926 . . . 4  |-  ( x  e.  ( _V  \  A )  <->  -.  x  e.  A )
43con2bii 333 . . 3  |-  ( x  e.  A  <->  -.  x  e.  ( _V  \  A
) )
51biantrur 508 . . 3  |-  ( -.  x  e.  ( _V 
\  A )  <->  ( x  e.  _V  /\  -.  x  e.  ( _V  \  A
) ) )
64, 5bitr2i 253 . 2  |-  ( ( x  e.  _V  /\  -.  x  e.  ( _V  \  A ) )  <-> 
x  e.  A )
76difeqri 3585 1  |-  ( _V 
\  ( _V  \  A ) )  =  A
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 370    = wceq 1437    e. wcel 1868   _Vcvv 3081    \ cdif 3433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-dif 3439
This theorem is referenced by:  dfun3  3711  dfin3  3712  invdif  3714  ssindif0  3846  difdifdir  3883
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