Equivalences are Interderivable/Proof 2

Theorem
If two propositional formulas are interderivable, they are equivalent:


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

Proof
Let $v$ be an arbitrary interpretation.

Then by definition of interderivable:
 * $v \left ({p \iff q}\right)$ iff $v \left ({p}\right) = v \left ({q}\right)$

Since $v$ is arbitrary, $v \left ({p}\right) = v \left ({q}\right)$ holds in all interpretations.

That is:
 * $p \dashv \vdash q$