Clearly, $\exists x \forall y \phi(x,y) \to \forall y \exists x \phi(x,y)$ is a tautology. However, the other way around is not a tautology: $\forall y \exists x \phi(x,y) \to \exists x \forall y \phi(x,y)$.
Nevertheless, I am interested in the case that this holds. That is, I am interested in the set of structures and formulae $\phi(x,y)$ where the following holds:
$$\exists x \forall y \phi(x,y) \leftrightarrow \forall y \exists x \phi(x,y)$$
Does there exist an analysis of structures where this statement holds? Are there interesting things to be said about them?
EDIT: Note that this does not require the statement to hold for arbitrary $\phi(x,y)$. i.e. if we need to restrict $\phi(x,y)$ in some way to get something interesting, then I still like to know about it.
Best Answer
I don't know if this answers your questions, but it's too long for a comment.
Put all of your of your statements $\phi(x,y)$ in a 2D array (I assumed countable statements, just for convenience):
\begin{array}{c c c c c} \phi(x_0,y_0) & \phi(x_1,y_0) & \phi(x_2,y_0) & \phi(x_3,y_0) & \dots\\ \phi(x_0,y_1) & \phi(x_1,y_1) & \phi(x_2,y_1) & \phi(x_3,y_1) & \dots \\ \phi(x_0,y_2) & \phi(x_1,y_2) & \phi(x_2,y_2) & \phi(x_3,y_2) & \dots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{array} then you'll notice that the first statement $\exists x\forall y$ is saying that a column in this array is correct. In the second case $\forall y \exists x$ you are simply saying that each row has at least one correct statement.
Thus the two are equal either if you only have $1$ column, or if there is at least one column where each row has a correct statement.