Incorrect Answer.


The semantic entailment x P(x) x Q(x) x (P(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) x Q(x) also satisfy x (P(x) Q(x)).

With these insights at hand, it is not hard to come up with a counterexample. Essentially, we only need to construct non-empty sets PM and QM such that their intersection is empty. So let the model M be given by

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

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