 Solution

A variable $v$ is fresh for the surrounding context $\Gamma$ when it does not occur free in any undischarged assumption in $\Gamma$. Choose it also distinct from bound variables that would cause variable capture in the intended substitution. The universal quantifier introduction rule isIts justification is that the logical derivation of $\varphi (v)$ used no special information about $v$, so it establishes the property of an arbitrary object. Semantically, changing the assignment of $v$ preserves the truth of $\Gamma$; therefore $\varphi (v)$ holds for every possible assigned object. Without the eigenvariable condition, the assumption $R(v)$ could incorrectly be generalized to $\forall xR(x)$ in an interpretation where only one of two objects satisfies $R$.