Biconditional is Associative

Theorem
Equivalence is associative:
 * $p \iff \left({q \iff r}\right) \dashv \vdash \left({p \iff q}\right) \iff r$

This can alternatively be rendered as:


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

These forms can be seen to be logically equivalent.

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

Proof by Truth Table
We apply the Method of Truth Tables.

As can be seen by inspection, the truth values under the main connective match for all models.

$\begin{array}{|ccccc||ccccc|} \hline p & \iff & (q & \iff & r) & (p & \iff & q) & \iff & r \\ \hline F & F & F & T & F & F & T & F & F & F \\ F & T & F & F & T & F & T & F & T & T \\ F & T & T & F & F & F & F & T & T & F \\ F & F & T & T & T & F & F & T & F & T \\ T & T & F & T & F & T & F & F & T & F \\ T & F & F & F & T & T & F & F & F & T \\ T & F & T & F & F & T & T & T & F & F \\ T & T & T & T & T & T & T & T & T & T \\ \hline \end{array}$