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Theorem brsymdif 25673
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 4213 . 2  |-  ( A ( R(++) S ) B  <->  <. A ,  B >.  e.  ( R(++) S
) )
2 elsymdif 25668 . . 3  |-  ( <. A ,  B >.  e.  ( R(++) S )  <->  -.  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  S ) )
3 df-br 4213 . . . 4  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 4213 . . . 4  |-  ( A S B  <->  <. A ,  B >.  e.  S )
53, 4bibi12i 307 . . 3  |-  ( ( A R B  <->  A S B )  <->  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  S ) )
62, 5xchbinxr 303 . 2  |-  ( <. A ,  B >.  e.  ( R(++) S )  <->  -.  ( A R B  <-> 
A S B ) )
71, 6bitri 241 1  |-  ( A ( R(++) S ) B  <->  -.  ( A R B  <->  A S B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    e. wcel 1725   <.cop 3817   class class class wbr 4212  (++)csymdif 25662
This theorem is referenced by:  brtxpsd  25739
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-dif 3323  df-un 3325  df-br 4213  df-symdif 25663
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