Theorem elsymdifxor 4226

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 4224	. 2 ⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ ¬ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶))
2		df-xor 1502	. 2 ⊢ ((𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶) ↔ ¬ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶))
3	1, 2	bitr4i 280	1 ⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ (𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶))

Syntax hints:  ¬ wn 3   ↔ wb 208   ⊻ wxo 1501   ∈ wcel 2114   △ csymdif 4218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-xor 1502  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3939  df-un 3941  df-symdif 4219

This theorem is referenced by:  dfsymdif2  4227  symdifass  4228