$ F(x(t-t_0))=e^{jwt_o}X(\omega) $
$ F(x(t-t_0))=e^{jwt_o}F(x(t)) $
$ F(x(t-t_0))=\int_{-\infty}^\infty x(t-t_0)e^{-jwt}dt $
Let $ \tau = t-t_0 $ and $ d\tau = dt $
$ F(x(t-t_0))=\int_{-\infty}^\infty x(\tau)e^{-jw(\tau+t_0)}dt $
$ F(x(t-t_0))=\int_{-\infty}^\infty x(\tau)e^{-jw\tau}e^{jwt_0}dt $ , $ e^{jwt_0} $ is a constant so,
$ F(x(t-t_0))=e^{jwt_0}\int_{-\infty}^\infty x(\tau)e^{-jw\tau}dt $
$ F(x(t-t_0))=e^{jwt_0} X(\omega) $
