Theorem dfsymdif2 3813

Description: Alternate definition of the symmetric difference. (Contributed by BJ, 30-Apr-2020.)

Assertion

Ref	Expression
dfsymdif2	⊢ (𝐴 △ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)}

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfsymdif2

Step	Hyp	Ref	Expression
1		elsymdifxor 3812	. 2 ⊢ (𝑥 ∈ (𝐴 △ 𝐵) ↔ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵))
2	1	abbi2i 2725	1 ⊢ (𝐴 △ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)}

Syntax hints:   ⊻ wxo 1456   = wceq 1475   ∈ wcel 1977  {cab 2596   △ 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: (None)