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 $$\mathcal{M} \left({p}\right) = \mathcal{M} \left({q}\right)$$.