NOR is Commutative/Proof by Truth Table

Theorem

$p \downarrow q \dashv \vdash q \downarrow p$

Proof

Apply the Method of Truth Tables:

$\begin{array}{|ccc||ccc|} \hline

p & \downarrow & q & q & \downarrow & p \\ \hline \F & \T & \F & \F & \T & \F \\ \F & \F & \T & \T & \F & \F \\ \T & \F & \F & \F & \F & \T \\ \T & \F & \T & \T & \F & \T \\ \hline \end{array}$

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

$\blacksquare$