Biconditional Equivalent to Biconditional of Negations/Formulation 1

Theorem

 * $p \iff q \dashv \vdash \neg p \iff \neg q$

This can be expressed as two separate theorems:

Proof
We apply the Method of Truth Tables.

As can be seen by inspection, in all cases the truth values under the main connectives match for all boolean interpretations.

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