Equivalences are Interderivable/Proof 1

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


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

Proof
The result follows directly from the truth table for the biconditional:

$\begin{array}{|cc||ccc|} \hline p & q & p & \iff & q \\ \hline \F & \F & \F & \T & \F \\ \F & \T & \F & \F & \T \\ \T & \F & \F & \F & \F \\ \T & \T & \F & \T & \T \\ \hline \end{array}$

By inspection, it is seen that $\map \MM {p \iff q} = \T$ $\map \MM p = \map \MM q$.