Equivalences are Interderivable

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


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

Proof by Natural Deduction
By the tableau method:

First, we show that if $p \dashv \vdash q$, then $p \iff q$:

Next, we show that if $p \iff q$, then $p \dashv \vdash q$:

Similarly:

Proof by Truth Table
The result follows directly from the truth table for material equivalence:

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

We see that $\mathcal M \left({p \iff q}\right) = T$ precisely when $\mathcal M \left({p}\right) = \mathcal M \left({q}\right)$.