I am trying to find a method to apply the implicit function theorem for subdifferential convex functions. The original theorem provides an equation for the partial derivative of the implicit function w.r.t $x$ by using the chain rule. There are some variants for non-differentiable functions but these only guarantee that a solution $x=x(y)$ to the equation $$ F\big(x(y),y\big) = 0 $$ exists and do not give any information about the partial derivative of $x$ respect to $y$. Is there any way to adapt this in terms of subdifferentials instead of partial derivatives?

Thanks