Biconditional Properties

Theorems
Equivalence is commutative:


 * $$p \iff q \dashv \vdash q \iff p$$

Equivalence is associative:


 * $$p \iff \left({q \iff r}\right) \dashv \vdash \left({p \iff q}\right) \iff r$$

Equivalence destroys copies of itself:


 * $$p \iff p \dashv \vdash \top$$

Proof by Natural deduction
Commutativity is proved by the Tableau method:

$$q \iff p \vdash p \iff q$$ is proved similarly.

Proof of associativity by natural deduction is just too tedious to be considered.

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