On Wigner's theorem in strictly convex normed spaces

In this note, we generalize the well-known Wigner's theorem. Let $X$ and $Y$ be real normed spaces and $Y$ strictly convex. We show that $f\colon X\to Y$ satisfies $\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\{\|x+y\|,\|x-y\|\}$, $x,y\in X$, if and only if $f$ is phase equivalent to a linear isometry.