Exclusive Or is Negation of Biconditional

Theorem

Exclusive or is equivalent to the negation of the biconditional:

$p \oplus q \dashv \vdash \neg \paren {p \iff q}$

Proof

 $\displaystyle p \oplus q$ $\dashv \vdash$ $\displaystyle \paren {p \lor q} \land \neg \paren {p \land q}$ Definition of Exclusive Or $\displaystyle$ $\dashv \vdash$ $\displaystyle \neg \paren {p \iff q}$ Non-Equivalence as Conjunction of Disjunction with Negation of Conjunction

$\blacksquare$