Theorem elsymdifxor 4180

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 ∈ (𝐵 △ 𝐶) ↔ (𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶))

Proof of Theorem elsymdifxor

Step	Hyp	Ref	Expression
1		elsymdif 4178	. 2 ⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ ¬ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶))
2		df-xor 1504	. 2 ⊢ ((𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶) ↔ ¬ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶))
3	1, 2	bitr4i 277	1 ⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ (𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶))

Syntax hints:  ¬ wn 3   ↔ wb 205   ⊻ wxo 1503   ∈ wcel 2108   △ csymdif 4172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-xor 1504  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-dif 3886  df-un 3888  df-symdif 4173

This theorem is referenced by:  dfsymdif2  4181  symdifass  4182