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
Let $$v: \left\{{p, q}\right\} \to \left\{{T, F}\right\}$$ be an interpretation for a logical formula $$\phi$$ of two variables $$p, q$$.

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