Are the quantifiers interchangeable?

Question

In other words, is it true that $\forall x \; \exists y\;\phi(x, y) \iff \exists y\;\forall x \; \phi(x, y) $?

Answer 1

Every child has a father, but there is not a father for all children.

Answer 2

$\forall x \in \mathbb{R} \exists y\in\mathbb{R}\,[x>y]$ is true, but $\exists y\in\mathbb{R}\forall x\in\mathbb{R}\,[x>y]$ is false.