How to prove the inequality of solutions for two Cauchy problems?

Question

Consider two continuous and Lipschitz continuous with respect to the second variable functions $g_1:[0,1]\times\mathbb{R}\to\mathbb{R}$ and $g_2:[0,1]\times\mathbb{R}\to\mathbb{R}$ such that $g_1\le g_2$. Also, $\phi_1$ and $\phi_2$ are functions defined on $[0,1]\times\mathbb{R}$ by $$ \begin{equation} \frac{\partial}{\partial t}\phi_i(t,x)=g_i(t,\phi_i(t,x))\\ \phi_i(0,x)=x\;. \end{equation} $$ How to prove that $\phi_1$ and $\phi_2$ are increasing in $x$ and $\phi_1\le\phi_2$? Thank you for your suggestion.