Three-Way Exclusive Or and Equivalence

Theorem
Let $$p \iff q$$ be the equivalence operation, and $$p \oplus q$$ be the exclusive or operation.

Then:
 * $$p \iff q \iff r \dashv \vdash p \oplus q \oplus r$$

Proof
From Equivalence Properties and Exclusive Or Properties, we have that both $$\iff$$ and $$\oplus$$ are associative, which justifies the rendition of this result without parentheses.

Proof by Truth Table
Let $$v: \left\{{p}\right\} \to \left\{{T, F}\right\}$$ be an interpretation for a boolean variable $$p$$.

We use the truth table results obtained in Equivalence Properties and Exclusive Or Properties.

Comment
A bizarrely non-intuitive result which is rarely documented.