Let $F:\mathcal{U\subset\mathbb{R}^3}\to \mathbb{R}$ of class $C'$, let $(t_0,x_0,y_0)\in \mathcal{U}$, such that $F(t_0,x_0,y_0)=0$ and $\partial_y F(t_0,x_0,y_0)\neq 0$. Prove that for all $\varepsilon>0$ there is a unique $\varphi:(t_0-\varepsilon,t_0+\varepsilon)\to\mathbb{R}$ such that $\varphi(t_0)=x_0$ and $\varphi'(t_0)=y_0$ with $F(t,\varphi(t),\varphi'(t))=0$ for all $t\in(t_0-\varepsilon,t_0+\varepsilon)$.
Comment: It seems to follow from the Implicit theorem, applying directly, I can find a function $\phi$ of class $C'$ in a neigborhood $N$ of $(t_0,x_0)$ such that $y=\phi(t,x)$ is a solution of $F(t,x,y)=0$ for all $(t,x)\in N$ and such that $\phi(t_0,x_0)=y_0$. But I could not relate the $\varphi'$ appearing in the statement. Any help?