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Theorem brsymdif 24757
Description: The binary relationship of a symmetric difference. (Contributed by Scott Fenton, 11-Apr-2012.)
Assertion
Ref Expression
brsymdif  |-  ( A ( R(++) S ) B  <->  -.  ( A R B  <->  A S B ) )

Proof of Theorem brsymdif
StepHypRef Expression
1 df-br 4061 . 2  |-  ( A ( R(++) S ) B  <->  <. A ,  B >.  e.  ( R(++) S
) )
2 elsymdif 24752 . . 3  |-  ( <. A ,  B >.  e.  ( R(++) S )  <->  -.  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  S ) )
3 df-br 4061 . . . 4  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 4061 . . . 4  |-  ( A S B  <->  <. A ,  B >.  e.  S )
53, 4bibi12i 306 . . 3  |-  ( ( A R B  <->  A S B )  <->  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  S ) )
62, 5xchbinxr 302 . 2  |-  ( <. A ,  B >.  e.  ( R(++) S )  <->  -.  ( A R B  <-> 
A S B ) )
71, 6bitri 240 1  |-  ( A ( R(++) S ) B  <->  -.  ( A R B  <->  A S B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    e. wcel 1701   <.cop 3677   class class class wbr 4060  (++)csymdif 24746
This theorem is referenced by:  brtxpsd  24819
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824  df-dif 3189  df-un 3191  df-br 4061  df-symdif 24747
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