How do I prove whether this formula is valid or satisfiable? $\exists xy[P(x,y) \rightarrow \forall xy\ P(x,y)]$ By valid/satisfiable I mean in the context of propositional logic. The following formula is:

$$\exists xy[P(x,y) \rightarrow \forall xy\ P(x,y)]$$

I understand that P is considered to be a predicate symbol, by P(x,y) is a formula that should evaluate to true or false. However, how do I go about solving this kind of problems if I don't have information on what x and y are?
By the rules I learned the sentence is not well formed, so it is neither valid nor satisfiable.  The square brackets mean the existential quantifier applies to everything inside, so you can't requantify over them with the universal quantifier.  It would be legal to write $\exists xy P(x,y) \implies \forall xy P(x,y)$ but this still creates an opportunity for confusion because of the reuse of $x,y$.  I find it better to write 
$$\exists xy P(x,y) \implies \forall wz P(w,z)$$
where the scope of the quantifiers is obvious.  I have seen authors who specifically allow reuse of variables, but they have to be very careful about the rules for substitution.  Some comments have said the rules they are used to accept your sentence, but it is equivalent to the one I set in display mode. 

This makes it easier to see that the sentence is satisfiable but not valid.  If $\forall wz P(w,z)$ is true the implication is true as well.  If $P(x,y)$ is true sometimes and false sometimes the implication is false.  We could easily construct a model for each of the previous sentences to prove our claim.
By the rules I learned the sentence is not well formed, so it is neither valid nor satisfiable.  The square brackets mean the existential quantifier applies to everything inside, so you can't requantify over them with the universal quantifier.  It would be legal to write $\exists xy P(x,y) \implies \forall xy P(x,y)$ but this still creates an opportunity for confusion because of the reuse of $x,y$.  I find it better to write 
$$\exists xy P(x,y) \implies \forall wz P(w,z)$$
where the scope of the quantifiers is obvious.  I have seen authors who specifically allow reuse of variables, but they have to be very careful about the rules for substitution.  Some comments have said the rules they are used to accept your sentence, but it is equivalent to the one I set in display mode. 

This makes it easier to see that the sentence is satisfiable but not valid.  If $\forall wz P(w,z)$ is true the implication is true as well.  If $P(x,y)$ is true sometimes and false sometimes the implication is false.  We could easily construct a model for each of the previous sentences to prove our claim.