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Theorem brsymdif 24372
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 4024 . 2  |-  ( A ( R(++) S ) B  <->  <. A ,  B >.  e.  ( R(++) S
) )
2 elsymdif 24367 . . 3  |-  ( <. A ,  B >.  e.  ( R(++) S )  <->  -.  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  S ) )
3 df-br 4024 . . . 4  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 4024 . . . 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 1684   <.cop 3643   class class class wbr 4023  (++)csymdif 24361
This theorem is referenced by:  brtxpsd  24434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-un 3157  df-br 4024  df-symdif 24362
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