De Morgan’s laws appear in set theory as $\overline{A \cup B} = \overline{A} \cap \overline{B}$ and in logic as $\lnot(p \lor q) = \lnot p \land \lnot q$. Why is one proof enough for both?

C. Because both are statements in the same Boolean algebra

One structure, two notations. This is why a database query optimiser, a circuit simplifier and a proof assistant can all apply the same rewrite rule.