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Theorem difdif 3460
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 546 . . 3  |-  ( x  e.  A  <->  ( x  e.  A  /\  (
x  e.  B  ->  x  e.  A )
) )
2 iman 414 . . . . 5  |-  ( ( x  e.  B  ->  x  e.  A )  <->  -.  ( x  e.  B  /\  -.  x  e.  A
) )
3 eldif 3317 . . . . 5  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
42, 3xchbinxr 303 . . . 4  |-  ( ( x  e.  B  ->  x  e.  A )  <->  -.  x  e.  ( B 
\  A ) )
54anbi2i 676 . . 3  |-  ( ( x  e.  A  /\  ( x  e.  B  ->  x  e.  A ) )  <->  ( x  e.  A  /\  -.  x  e.  ( B  \  A
) ) )
61, 5bitr2i 242 . 2  |-  ( ( x  e.  A  /\  -.  x  e.  ( B  \  A ) )  <-> 
x  e.  A )
76difeqri 3454 1  |-  ( A 
\  ( B  \  A ) )  =  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    \ cdif 3304
This theorem is referenced by:  dif0  3685  undifabs  3692
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 2411
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 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-v 2945  df-dif 3310
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