NAND as Disjunction of Negations/Proof by Truth Table

Proof
We apply the Method of Truth Tables.

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

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