Are these two predicate statements equivalent or not? $\exists x \forall y P(x,y) \equiv \forall y \exists x P(x,y)$
I was told they were not, but I don't see how it can be true.

Take the more concrete statements: $\forall {n \in \mathbb{N}}\ \exists {m \in \mathbb{N}}: m > n$ versus $\exists {m \in \mathbb{N}}\ \forall {n \in \mathbb{N}}: m > n$. Now the first statement reads: For every natural number there's a bigger one. The second statement reads: There's a biggest natural number.
Quantifiers of different type don't commute. However, quantifiers of the same type do: $\exists x \ \exists y \equiv \exists y \ \exists x$ and $\forall x \ \forall y \equiv \forall y \ \forall x$.
