Theorem dfsymdif4 4182

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 4181	. 2 ⊢ (𝑥 ∈ (𝐴 △ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2	1	abbi2i 2879	1 ⊢ (𝐴 △ 𝐵) = {𝑥 ∣ ¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)}

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1539   ∈ wcel 2106  {cab 2715   △ csymdif 4175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-un 3892  df-symdif 4176

This theorem is referenced by:  mbfeqalem1  24805