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Theorem brsymdif 4462
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 4406 . 2  |-  ( A ( R  /_\  S ) B  <->  <. A ,  B >.  e.  ( R  /_\  S ) )
2 elsymdif 3670 . . 3  |-  ( <. A ,  B >.  e.  ( R  /_\  S )  <->  -.  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  S ) )
3 df-br 4406 . . . 4  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 4406 . . . 4  |-  ( A S B  <->  <. A ,  B >.  e.  S )
53, 4bibi12i 317 . . 3  |-  ( ( A R B  <->  A S B )  <->  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  S ) )
62, 5xchbinxr 313 . 2  |-  ( <. A ,  B >.  e.  ( R  /_\  S )  <->  -.  ( A R B  <-> 
A S B ) )
71, 6bitri 253 1  |-  ( A ( R  /_\  S ) B  <->  -.  ( A R B  <->  A S B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188    e. wcel 1889    /_\ csymdif 3664   <.cop 3976   class class class wbr 4405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-v 3049  df-dif 3409  df-un 3411  df-symdif 3665  df-br 4406
This theorem is referenced by:  brtxpsd  30673
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