Unions of two elementary classes that have special properties

Question

This is a follow up to my previous question on the union of two elementary classes. Define an $\forall$-elementary class as a class $K$ of structures in some fixed language $L$ that can be axiomatized by $\forall$-sentences. An $\exists$-elementary class is defined analogously. It is not too hard to show that the intersection of two $\forall$-elementary classes is itself $\forall$-elementary, and similarly for $\exists$-elementary. What about the union? Is the union of two $\forall$-elementary classes (respectively $\exists$-elementary classes) itself $\forall$-elementary (respectively $\exists$-elementary)?

Answer 1

The disjunction of $\exists$-sentences is equivalent to an $\exists$-sentence and the disjunction of $\forall$-sentences is equivalent to a $\forall$-sentence. So applying the argument of the answer to your previous question, both of the questions here have affirmative answers.

The closure of $\exists$-sentences under disjunction is immediate since $\exists$ distributes over $\vee$. For the $\forall$-case, we use a cute trick: $$[\forall x(P(x))]\vee[\forall x(Q(x))]$$ is equivalent to $$\forall x,y[P(x)\vee Q(y)],$$ which is a $\forall$-sentence.