Techniques for Reversing the Order of Quantifiers

It is well-known that in general, the order of universal and existential quantifiers cannot be reversed. In other words, for a general logical formula $\phi(\cdot,\cdot)$, $(\forall x)(\exists y) \phi(x,y) \quad \not\Leftrightarrow \quad (\exists y)(\forall x) \phi(x,y)$ On the other hand, we know the right-hand side is more restrictive than the left-hand side; that is, $(\exists y)(\forall x) \phi(x,y) \Rightarrow (\forall x)(\exists y) \phi(x,y)$. This question focuses on techniques to derive $(\forall x)(\exists y) \phi(x,y) \Rightarrow (\exists y)(\forall x) \phi(x,y)$, whenever it holds for $\phi(\cdot,\cdot)$. Diagonalization is one such technique. I first see this use of diagonalization in the paper Relativizations of the $\mathcal{P} \overset{?}{=} \mathcal{NP}$ Question (see also the short note by Katz ). In that paper, the authors first prove that: For any deterministic, polynomial-time oracle machine M, there exists a language B such that $L_B \ne L(M^B)$. They then reverse the order of the quantifiers (using diagonalization ), to prove that: There exists a language B such that for all deterministic, poly-time M we have $L_B \ne L(M^B)$. This technique is used in other papers, such as [CGH] and [AH]. I found another technique in the proof of Theorem 6.3 of [IR]. It uses a combination of measure theory and pigeon-hole principle to reverse the order of quantifiers. I want to know what other techniques are used in computer science, to reverse the order of universal and existential quantifiers?