### Correct 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).

• A model M satisfies x (P(x) Q(x)) if, and only if the set PM is contained in the set QM.
• A model M satisfies x P(x)  x Q(x) if, and only if the set QMequals the model's set of values, A, provided that PMequals the model's set of values, A, to begin with.
So let M be any model with M  x (P(x) Q(x)). According to the above, this means that the set PM is contained in the set QM. We need to show that M  x P(x)  x Q(x) holds as well. Since the latter formula is an implication, this amounts to showing that M  x Q(x), assuming that M  x P(x). But the latter assumption means that the set PM equals A, the set of values of M. However, we already established (where?) that PM is contained in QM, which is contained in A by definition. Thus, A is contained in QM and vice versa. Thus, QM equals A, meaning that M  x Q(x) holds.

Therefore, the semantic entailment above is indeed valid in predicate logic.
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