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 ∈ (𝐵 △ 𝐶) ↔ (𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶))

Proof of Theorem elsymdifxor

Step	Hyp	Ref	Expression
1		xnor 1458	. . 3 ⊢ ((𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶) ↔ ¬ (𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶))
2	1	notbii 309	. 2 ⊢ (¬ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶) ↔ ¬ ¬ (𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶))
3		elsymdif 3811	. 2 ⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ ¬ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶))
4		notnotb 303	. 2 ⊢ ((𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶) ↔ ¬ ¬ (𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶))
5	2, 3, 4	3bitr4i 291	1 ⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ (𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶))

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