Given a knot $K$ in the three sphere, we compare the knot Floer homology of $(S3, K)$ with the knot Floer homology of $(\Sigma(K), K)$, where $\Sigma(K)$ is the double branched cover of the three-sphere over $K$. By studying an involution on the symmetric product of a Heegaard surface for $(\Sigma(K), K)$ whose fixed set is a symmetric product of a Heegaard surface for $(S3, K)$, and applying recent work of Seidel and Smith, we produce an analog of the classical Smith inequality for cohomology for knot Floer homology. To wit, we show that the rank of the knot Floer homology of $(S3,K)$ is less than or equal to the rank of the knot Floer homology of $(\Sigma(K), K)$