A compiler rewrites if (!(a && b)) as if (!a || !b). Which law licenses that, and why does the compiler bother?

D. De Morgan's law, and the rewritten form can short-circuit on !a

The truth tables match, so the substitution is safe, and the second form reaches a decision on !a alone where the first must evaluate the whole conjunction before negating. Equivalence is what licenses it and short-circuiting is what makes it worth doing, which is the same pairing as in logic minimisation.