Proof using Banach-Steinhaus theorem

Let $\phi_n:X \rightarrow \mathbb{R}$ be a sequence $(n=0,1,\dots)$ of linear functionals on X, where X is Banach and for which the map $\Phi:X\ni x \longrightarrow(\phi_n(x))_{n=0}^{\infty}\in \ell^1$ is well defined. Prove that:$$\Phi \text{ is continous } \iff\phi_n \text{ is continuous for any } n\ge0$$

My attempt

Proof from left to right is obvious. From right to left I know I should use Banach Steinhaus theorem but I can't see how it helps.