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Theorem brsymdif 4477
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 4421 . 2  |-  ( A ( R  /_\  S ) B  <->  <. A ,  B >.  e.  ( R  /_\  S ) )
2 elsymdif 3698 . . 3  |-  ( <. A ,  B >.  e.  ( R  /_\  S )  <->  -.  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  S ) )
3 df-br 4421 . . . 4  |-  ( A R B  <->  <. A ,  B >.  e.  R )
4 df-br 4421 . . . 4  |-  ( A S B  <->  <. A ,  B >.  e.  S )
53, 4bibi12i 316 . . 3  |-  ( ( A R B  <->  A S B )  <->  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  S ) )
62, 5xchbinxr 312 . 2  |-  ( <. A ,  B >.  e.  ( R  /_\  S )  <->  -.  ( A R B  <-> 
A S B ) )
71, 6bitri 252 1  |-  ( A ( R  /_\  S ) B  <->  -.  ( A R B  <->  A S B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    e. wcel 1868    /_\ csymdif 3692   <.cop 4002   class class class wbr 4420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-dif 3439  df-un 3441  df-symdif 3693  df-br 4421
This theorem is referenced by:  brtxpsd  30654
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