Let $T$ be $\Sigma_1$ complete theory. I could understand the proof that 1-consistency of $T$ implies $\Sigma_1$ soundness of $T$, but I couldn't vice versa.
I wonder if there are some missing hypotheses about $T$ because the book now I'm referencing is a casual/informal/not strict book.
Assume $T$ is $\Sigma_1$ sound and that $T\vdash\exists x\varphi(x)$, where $\varphi$ is $\Delta_0$. Since $T$ is $\Sigma_1$-sound, $\exists x\varphi(x)$ is true, and thus $\varphi(m)$ is true for some $m\in\mathbb N.$ Then by $\Sigma_1$-completeness, $T\vdash \varphi(\bf m).$ So $T$ is 1-consistent.