Theorem dfsymdif4 4239

Description: Alternate definition of the symmetric difference. (Contributed by NM, 17-Aug-2004.) (Revised by AV, 17-Aug-2022.)

Assertion

Ref	Expression
dfsymdif4	⊢ (𝐴 △ 𝐵) = {𝑥 ∣ ¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)}

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfsymdif4

Step	Hyp	Ref	Expression
1		elsymdif 4238	. 2 ⊢ (𝑥 ∈ (𝐴 △ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2	1	eqabi 2871	1 ⊢ (𝐴 △ 𝐵) = {𝑥 ∣ ¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)}

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1540   ∈ wcel 2109  {cab 2714   △ csymdif 4232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-dif 3934  df-un 3936  df-symdif 4233

This theorem is referenced by:  mbfeqalem1  25599