Equivalences are Interderivable/Proof 2

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


 * $\paren {p \dashv \vdash q} \dashv \vdash \paren {p \iff q}$

Proof
Let $v$ be an arbitrary interpretation.

Then by definition of interderivable:
 * $\map v {p \iff q}$ $\map v p = \map v q$

Since $v$ is arbitrary, $\map v p = \map v q$ holds in all interpretations.

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