Proof of equivalence of solutions to initial value problem given same gradient

Question

I have two functions $x(t)$ & $y(t)$ I have $x(0) = y(0)$ and that $\dot{x} = f(x,y)$ & $\dot{y} = g(x,y)$ I have that when $x = y$, $\dot{x} = \dot{y}$.

I want to show that $x(t) = y(t)$, I don't know how to formalise this however.