I'm very familiar with boolean quantifiers when written at the beginning of an expression like so:
$$ \forall x: \exists y: P(x,y) $$
However I've also seen the convention to write boolean quantifiers at the end. My question is do these conventions translate to one another by shifting like this:
$$ \forall x: \exists y: P(x,y)\quad \equiv\quad P(x,y)\quad\forall x\ \exists y $$
or by reflection like this:
$$ \forall x: \exists y: P(x,y)\quad \equiv\quad P(x,y)\quad \exists y\ \forall x\ ? $$
I have never seen an example where there are multiple quantifiers on the right side. The only such usage I am aware of is using one "$\forall$" at the end of a expression. $$f(x) = 0 \, \forall x.$$
This usually done to make things easier to read out loud. Here an example:
$$\forall \epsilon >0 :\exists \delta >0 :\forall x \in B_{x_0}(\delta) : |f(x)-f(y)| \leq \epsilon$$
This could be read as: "For an arbitrary positive number epsilon there is a number delta such that $|f(x) - f(x_0)| \leq \epsilon$ for all $y$ within $\delta$ of $x_0$."
So you could write this as "...$|f(x)-f(x_0)| \leq \epsilon \forall y \in B_{x_0}(\delta)$"
So this appended "$\forall$" is to be read as the last quator before the expression if you want to stick to the technical rules.
PS: I'm not justifying this notation, I just want to explain how it is usually used.