As Emil Jeřábek and James Hanson mentioned in comments, this is well-known in the literature of first-order arithmetic as the $\Sigma_1$ least number principle, $\mathsf{L}\Sigma_1$. Simpson doesn't mention it in his book (he doesn't really talk about $\Sigma_n$ bounding either, which is more well known).
Over $\mathsf{PA}^- + \mathsf{I}\Sigma_0$, Kirby and Paris [1] proved that $\mathsf{L}\Sigma_n$, $\mathsf{L}\Pi_n$, $\mathsf{I}\Sigma_n$, $\mathsf{I}\Pi_n$ are all equivalent.
Over $\mathsf{PA}^- + \mathsf{I}\Sigma_0 + \mathsf{exp}$, Slaman [2] proved (building on results of Kirby and Paris) that $\mathsf{L}\Delta_n$, $\mathsf{I}\Delta_n$, $\mathsf{B}\Sigma_n$, are all equivalent.
Both reversals hold over $\mathsf{RCA}_0$, since it proves $\mathsf{PA}^- + \mathsf{I}\Sigma_0 + \mathsf{exp}$. In particular, $\mathsf{L}\Sigma_1$ holds in $\mathsf{RCA}_0$ (as Emil Jeřábek mentioned).
[1] L. A. S. Kirby and J. B. Paris. Initial segments of models of Peano’s axioms. Set theory and hierarchy theory, V (Proc. Third Conf., Bierutowice, 1976), Springer, Berlin, 1977, pp. 211–226. Lecture Notes in Math., Vol. 619.
[2] Theodore A. Slaman. $\Sigma_n$-bounding and $\Delta_n$-induction. Proceedings of the American Mathematical Society 132(8), pp. 2449-2456, 2004.