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Theorem elsymdifxor 3721
Description: Membership in a symmetric difference is an exclusive-or relationship. (Contributed by David A. Wheeler, 26-Apr-2020.)
Assertion
Ref Expression
elsymdifxor  |-  ( A  e.  ( B  /_\  C )  <-> 
( A  e.  B  \/_  A  e.  C ) )

Proof of Theorem elsymdifxor
StepHypRef Expression
1 xnor 1364 . . 3  |-  ( ( A  e.  B  <->  A  e.  C )  <->  -.  ( A  e.  B  \/_  A  e.  C )
)
21notbii 294 . 2  |-  ( -.  ( A  e.  B  <->  A  e.  C )  <->  -.  -.  ( A  e.  B  \/_  A  e.  C )
)
3 elsymdif 3720 . 2  |-  ( A  e.  ( B  /_\  C )  <->  -.  ( A  e.  B  <->  A  e.  C ) )
4 notnot 289 . 2  |-  ( ( A  e.  B  \/_  A  e.  C )  <->  -. 
-.  ( A  e.  B  \/_  A  e.  C ) )
52, 3, 43bitr4i 277 1  |-  ( A  e.  ( B  /_\  C )  <-> 
( A  e.  B  \/_  A  e.  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/_ wxo 1362    e. wcel 1823    /_\ csymdif 3714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-xor 1363  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-dif 3464  df-un 3466  df-symdif 3715
This theorem is referenced by:  dfsymdif2  3722
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