Incorrect Answer.


The semantic entailment x (P(x) Q(x)) x P(x) x Q(x) is valid in predicate logic if, and only if all models M (that have relations PM and QM defined) that satisfy x (P(x) Q(x)) also satisfy x P(x) x Q(x).

With these insights at hand, it is not hard to come up with a counterexample. Essentially, we only need to ensure that either PM or QM equals A, and that their intersection is different from A. So let the model M be given by

A = {a,b};
PM = {a,b}
QM = {b}.

Please verify that we have M x (P(x) Q(x)), but that we do not have M x P(x) x Q(x). Thus, the semantic entailment above is not valid in predicate logic.
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