This is Cerf's pseudoisotopy-implies-isotopy theorem.
Cerf's result is true in high dimensions, while it's independently known in a low-dimensional range. In dimension $2$ it goes back to the Earle–Eells result that $\pi_0\, \hbox{Diff}(S^2)$ has precisely two components, and in dimension $3$ it is the analogous theorem of Cerf. But I think it's an open problem for $n=4$.
Jean Cerf (1970), La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes. Etudes Sci Publ. Math 39:5–173.
