Predicate Logic - Does order matter with Universal or existential

Question

So for example if i have $(\exists y\forall x(P(x) ∨ Q(y)) \implies \forall x\exists y(P(x) \vee Q(y)))$ Is this true or false?

The $\implies$ means logically implies, which is different than implies ($\to$). Logically implies means that for every thing that satisfies the first part, the second part is also satisfied by the same thing.

Answer 1

$$\forall x\exists y(x+x=y)$$ means just that we can always find $2x$.

$$\exists y\forall x(x+x=y)$$ means something quite different - that there is some $y$ such that $x+x=y$ for every $x$.

Answer 2

In general $\exists y \forall x\, R(x,y) \implies \forall x \exists y\, R(x,y)$

If there is something related to all things, then everything has something which is related to it.

However, if everything has something which is related to it, it may not be so that there's a same thing related to everything.