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?