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Theorem difdif 3315
Description: Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
Assertion
Ref Expression
difdif  |-  ( A 
\  ( B  \  A ) )  =  A

Proof of Theorem difdif
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pm4.45im 545 . . 3  |-  ( x  e.  A  <->  ( x  e.  A  /\  (
x  e.  B  ->  x  e.  A )
) )
2 iman 413 . . . . 5  |-  ( ( x  e.  B  ->  x  e.  A )  <->  -.  ( x  e.  B  /\  -.  x  e.  A
) )
3 eldif 3175 . . . . 5  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
42, 3xchbinxr 302 . . . 4  |-  ( ( x  e.  B  ->  x  e.  A )  <->  -.  x  e.  ( B 
\  A ) )
54anbi2i 675 . . 3  |-  ( ( x  e.  A  /\  ( x  e.  B  ->  x  e.  A ) )  <->  ( x  e.  A  /\  -.  x  e.  ( B  \  A
) ) )
61, 5bitr2i 241 . 2  |-  ( ( x  e.  A  /\  -.  x  e.  ( B  \  A ) )  <-> 
x  e.  A )
76difeqri 3309 1  |-  ( A 
\  ( B  \  A ) )  =  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    \ cdif 3162
This theorem is referenced by:  dif0  3537  undifabs  3544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-dif 3168
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