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Theorem elsymdifxor 4014
 Description: Membership in a symmetric difference is an exclusive-or relationship. (Contributed by David A. Wheeler, 26-Apr-2020.) (Proof shortened by BJ, 13-Aug-2022.)
Assertion
Ref Expression
elsymdifxor (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))

Proof of Theorem elsymdifxor
StepHypRef Expression
1 elsymdif 4012 . 2 (𝐴 ∈ (𝐵𝐶) ↔ ¬ (𝐴𝐵𝐴𝐶))
2 df-xor 1634 . 2 ((𝐴𝐵𝐴𝐶) ↔ ¬ (𝐴𝐵𝐴𝐶))
31, 2bitr4i 269 1 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 197   ⊻ wxo 1633   ∈ wcel 2155   △ csymdif 4006 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743 This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-xor 1634  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-dif 3737  df-un 3739  df-symdif 4007 This theorem is referenced by:  dfsymdif2  4015  symdifass  4016
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