I'm currently working my way through the details of first-order logic (using Suppes' Introduction to Logic), and I have a question about universal specification (US) (aka universal instantiation or elimination). Specifically, my question has to do with the choosing of a name for the 'ambiguous object' that is used to replace the dropped universally quantified variable (in Suppes' text, he uses the nomenclature 'ambiguous object' when applying either US or ES). Even more specifically, my question is what, if any, restriction applies to choosing of names of ambiguous objects when dropping / replacing multiple universally quantified variables.
For example, I have no trouble in grappling with an application of US in which the sentence contains but a single quantified variable. For example, given the following sentence, where $P$ is a 1-place predicate: $(\forall x)(Px)$
And after applying US, choosing $\alpha$ as the ambiguous object (the thing being instantiated), we get: $P\alpha$. This makes complete sense to me.
But what if we have the following sentence (in which $T$ is a 2-place predicate): $(\forall x)(\forall y)(\forall z)((xTy \bigwedge yTz) \to xTz)$
It's unclear to me if the following application of US is permitted or not:
$((\alpha T\alpha \bigwedge \alpha T\alpha) \to \alpha T\alpha)$
In Suppes' text, I have not come across an example in which an application of US such as this is used. However, I also cannot find mention of any sort of restriction on US that would not permit this. Another way to articulate my question is, given the application of US on just $x$, we get the following:
$(\forall y)(\forall z)((\alpha Ty \bigwedge yTz) \to \alpha Tz)$
Is there any such restriction on US that would prevent its application on $y$ such that $\alpha$ cannot legally be used as the object being instantiated? (For what it's worth, I do understand that $\alpha$ cannot be used if $y$ was existentially quantified, and we wanted to apply ES; I'm unclear if US has the same restriction.)
Thank you in advance for your time and help.