Past exam of the logic and set theory course of the University of Oxford / 2009 / A / 1 / c / Solution

Take $\alpha =\beta =P(x)$ for a unary predicate symbol. The premise $\forall x(P(x)\to P(x))$ is valid, but the proposed conclusion is $P(x)\to \forall xP(x)$. On a domain $\{0,1\}$ with $P=\{0\}$ and assignment $x\mapsto 0$, its antecedent is true and its consequent false.

Hence dropping the condition that $x$ is not free in $\alpha$ would turn a valid premise into an invalid conclusion. The condition permits the antecedent to remain fixed while the quantified variable ranges over the domain; an antecedent depending on that variable cannot be held true for all its values merely because it is true at the current assignment.