In preparing for an exam, I'm working through old exam questions and am now trying to figure out if the following first-order formula is valid. FYI, $m$ is a binary predicate.
$$(\forall x \,\exists y: m(x,y)) \implies (\exists y \, \forall x: m(x,y))$$
If I were to turn the original formula into English, I would say: "IF for all $x$ and some $y$, $m(x,y)$ is true, THEN for all $x$ and some $y$, $m(x,y)$ is true." This seems trivially true and I'm wondering if the order of the qualifiers matters and I'm reading the formula incorrectly?
Digging deeper, I reduced the given formula to this:
$$(\exists x\, \forall y: m(x,y)) \vee (\exists y \, \forall x: m(x,y))$$
In English this is: "For all $y$ and some $x$, $m(x,y)$ is true" OR "For all $x$ and some $y$, $m(x,y)$ is true." Which seems valid to me, though it would helpful to know the actual predicate formula of $m(x,y)$. It also seems to be making a somewhat different statement than the original formula so either I've interpreted the original formula incorrectly, I've reduced it to this one incorrectly, or I'm interpreting this one incorrectly.
Can anyone shed some light if I have the right thought process? Thank so much for the help!