Equivalences are Interderivable/Proof 1

Theorem
If two statement forms are interderivable, they are equivalent:


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

Proof
The result follows directly from the truth table for the biconditional:

$\begin{array}{|cc||ccc|} \hline p & q & p & \iff & q \\ \hline F & F & F & T & F \\ F & T & F & F & T \\ T & F & F & F & F \\ T & T & F & T & T \\ \hline \end{array}$

By inspection, it is seen that $\mathcal M \left({p \iff q}\right) = T$ precisely when $\mathcal M \left({p}\right) = \mathcal M \left({q}\right)$.