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Theorem brsymdif 4452
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 4396 . 2  |-  ( A ( R  /_\  S ) B  <->  <. A ,  B >.  e.  ( R  /_\  S ) )
2 elsymdif 3659 . . 3  |-  ( <. A ,  B >.  e.  ( R  /_\  S )  <->  -.  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  S ) )
3 df-br 4396 . . . 4  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 4396 . . . 4  |-  ( A S B  <->  <. A ,  B >.  e.  S )
53, 4bibi12i 322 . . 3  |-  ( ( A R B  <->  A S B )  <->  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  S ) )
62, 5xchbinxr 318 . 2  |-  ( <. A ,  B >.  e.  ( R  /_\  S )  <->  -.  ( A R B  <-> 
A S B ) )
71, 6bitri 257 1  |-  ( A ( R  /_\  S ) B  <->  -.  ( A R B  <->  A S B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189    e. wcel 1904    /_\ csymdif 3653   <.cop 3965   class class class wbr 4395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-dif 3393  df-un 3395  df-symdif 3654  df-br 4396
This theorem is referenced by:  brtxpsd  30732
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