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:

$\map v {p \iff q}$ if and only if $\map v p = \map v q$

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

That is:

$p \dashv \vdash q$

$\blacksquare$