After comparing conditional forms of quantified implications, we now collect general logical laws for quantified statements. These laws explain when quantifiers distribute over connectives and when they do not. Some forms are equivalences, while others are only one-way implications. In this lesson, logical laws for quantified statements will be tested through formulas, proof, and counterexamples.

For predicates $p(x)$ and $q(x)$, $$\begin{equation*} \exists x[p(x)\land q(x)]\to[(\exists x\,p(x))\land(\exists x\,q(x))] \end{equation*}$$ is true. The converse is not always true.