I was able to prove the statement $\forall{x} \forall{y} P(x,y) \vdash \forall{x} \forall{y} P(y,x)$:
This implies order does not matter within predicates. However, this confuses me because depending on what the predicate represents, order can be crucial.
Take, for example $P(x,y)$ representing $x < y$ with the universe of discourse for $x$ being all negative integers and the UOD for y being all positive integers.
In this case, $\forall{x} \forall{y} P(x,y)$ holds since all negative integers are smaller than all positive integers.
However, $\forall{x} \forall{y} P(y,x)$ would not hold since all positive integers are not smaller than all negative integers.
Clearly, I am misunderstanding something or making incorrect assumptions about the nature of predicate logic. Any clarification would be greatly appreciated.
It is a general principle that $\forall x \forall y \ P(x,y)$ is logically equivalent to $\forall x \forall y \ P(y,x)$, and that is what you proved (well, technically, you showed that $\forall x \forall y \ P(x,y)$ logically implies $\forall x \forall y \ P(y,x)$, but a completlely analogous proof will show that it goes the other way around as well)
However, it is understood here that $x$ and $y$ come from the same domain. Indeed, the very use of $\forall x \forall y$ ... implies that $x$ and $y$ come from the same domain.
Now, if you want $x$ and $y$ to be from different domains, then there are a few options.
First, if you are allowed the use of restricted quantifiers, then you can do $\forall x \in X \forall y \in Y ....$, ... but there is no general principle that says that $\forall x \in X \forall y \in Y \ P(x,y)$ is equivalent to $\forall x \in X \forall y \in Y \ P(y,x)$ (indeed, as your example demonstrates, we better not make this an euivelence principle)
Second, if you are not allowed the use of restricted quantifiers, you can do: $\forall x \forall y ((x \in X \land y \in Y) \to P(x,y))$, but now note that with the general principle that would be equivalent to $\forall x \forall y ((y \in X \land x \in Y) \to P(y,x))$ ... and that is not the same as $\forall x \forall y ((x \in X \land y \in Y) \to P(y,x))$
Finally, maybe you have learned logic in a way where you can just stipulate that 'variable $x$ comes from this domain, and variable $y$ from this other'. Well, you might get away with that in some informal setting, but given the hard principles of formal logic, the use of $\forall x \forall y ...$ still means that $x$ and $y$ must be from the same domain.
So no, your example with positive and negative numbers does not provide an exception to the equivalence principle or to your proof.