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Theorem difdif 3304
Description: Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
Assertion
Ref Expression
difdif  |-  ( A 
\  ( B  \  A ) )  =  A
Dummy variable  x is distinct from all other variables.

Proof of Theorem difdif
StepHypRef Expression
1 pm4.45im 547 . . 3  |-  ( x  e.  A  <->  ( x  e.  A  /\  (
x  e.  B  ->  x  e.  A )
) )
2 iman 415 . . . . 5  |-  ( ( x  e.  B  ->  x  e.  A )  <->  -.  ( x  e.  B  /\  -.  x  e.  A
) )
3 eldif 3164 . . . . 5  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
42, 3xchbinxr 304 . . . 4  |-  ( ( x  e.  B  ->  x  e.  A )  <->  -.  x  e.  ( B 
\  A ) )
54anbi2i 677 . . 3  |-  ( ( x  e.  A  /\  ( x  e.  B  ->  x  e.  A ) )  <->  ( x  e.  A  /\  -.  x  e.  ( B  \  A
) ) )
61, 5bitr2i 243 . 2  |-  ( ( x  e.  A  /\  -.  x  e.  ( B  \  A ) )  <-> 
x  e.  A )
76difeqri 3298 1  |-  ( A 
\  ( B  \  A ) )  =  A
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685    \ cdif 3151
This theorem is referenced by:  dif0  3526  undifabs  3533
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-v 2792  df-dif 3157
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