In set theory, the axiom of specification says that $\forall x_0\exists x_1\forall x_2\left(x_2\in x_1\leftrightarrow x_2\in x_0\land\theta\left[x_2\right]\right)$, where $\theta\left[x_2\right]$ is any formula that has $x_2$ as the only free variable.

In first order logic, the rule of universal introduction says that if $\Sigma\vdash\phi\left[t\right]$ then $\Sigma\vdash\forall x\phi\left[x/t\right]$, where $\Sigma$ is a set of axioms, $\phi$ is any formula that has $x$ as the only free variable and $t$ is any term. There is one restriction: $t$ cannot appear in $\Sigma$.

The problem is that, in set theory, every instance of the axiom of specification belong to $\Sigma$; in particular, instances of the axiom of specification in which a term appears belong to $\Sigma$. So every term appears in $\Sigma$, and you cannot use the rule of universal introduction at all. Without that rule, no relevant proof can be written down as far as I know.

I'm not sure if the question is clear enough; please let me know if it isn't.

Thanks.