Theorem dfsymdif4 4205

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 4204	. 2 ⊢ (𝑥 ∈ (𝐴 △ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2	1	eqabi 2895	1 ⊢ (𝐴 △ 𝐵) = {𝑥 ∣ ¬ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)}

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570   ∈ wcel 2145  {cab 2738   △ csymdif 4198

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 3902  df-un 3904  df-symdif 4199

This theorem is used by:  mbfeqalem1  25901