Proving that the elements of a free group satisfy a certain property

Question

Let $X$ be a set and $F(X)$ the free group on $X$. Suppose $P(x)$ denotes some property. If I want to show that all the elements of $x\in F(X)$ satisfy $P(x)$, would it be enough to show that $P(x)$ holds for all reduced words $x$? It seems to me that this is not true. For example let $\alpha\ne\beta\in X$. Then $\alpha\alpha\beta$ is not a reduced word of $F(X)$. So the suggested approach couldn't have covered $P(\alpha\alpha\beta)$. Is this correct?