Biconditional Equivalent to Biconditional of Negations/Formulation 1

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Theorem

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


This can be expressed as two separate theorems:

Forward Implication

$p \iff q \vdash \neg p \iff \neg q$

Reverse Implication

$\neg p \iff \neg q \vdash p \iff q$


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.

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

$\blacksquare$