I'm trying to derive the following as a theorem in a FOL Hilbert system:
$$∀x∃yQ(xy) → ∃yQ(yy) $$
But I'm a bit confused as to how one should interpret the incidence of two variables in $Q$. Is there variable collision or are the two statements separated by the logical implication not related in that sense? Is this an instantiation?
Beside the points stated in the comments, if you are not somehow familiar with the formula, you ought to check its validity, whether there is any counter-instance to it, preferably, by a semantic tableaux method.
A counter-instance has to make $\forall x\exists yQxy$ true and $\exists yQyy$ false. Eventually, you will see that in a domain with one element $\{a\}$ the formula does not have a counter-instance, but in domains with two $\{a, b\}$ (or more) elements, it does (the tableau will not close): There exists an assignment of truth values which verifies $Qab$ and $Qba$, but falsifies $Qaa$ and $Qbb$. Hence, it is not a valid formula.
Just posted by a different user: Naming random variables.
The point is that while it is technically correct to recycle a variable among different scopes (in the given sentence, each quantifier's scope does not extend across the material conditional "$→$"), many dislike this practice for poor communication and being misleading.
Christian's answer here is careful about not recycling variable names, whereas mine isn't so much.
