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Theorem elsymdifxor 3812
 Description: Membership in a symmetric difference is an exclusive-or relationship. (Contributed by David A. Wheeler, 26-Apr-2020.)
Assertion
Ref Expression
elsymdifxor (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))

Proof of Theorem elsymdifxor
StepHypRef Expression
1 xnor 1458 . . 3 ((𝐴𝐵𝐴𝐶) ↔ ¬ (𝐴𝐵𝐴𝐶))
21notbii 309 . 2 (¬ (𝐴𝐵𝐴𝐶) ↔ ¬ ¬ (𝐴𝐵𝐴𝐶))
3 elsymdif 3811 . 2 (𝐴 ∈ (𝐵𝐶) ↔ ¬ (𝐴𝐵𝐴𝐶))
4 notnotb 303 . 2 ((𝐴𝐵𝐴𝐶) ↔ ¬ ¬ (𝐴𝐵𝐴𝐶))
52, 3, 43bitr4i 291 1 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ⊻ wxo 1456   ∈ wcel 1977   △ csymdif 3805 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-xor 1457  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-symdif 3806 This theorem is referenced by:  dfsymdif2  3813
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