That is how science works - you have to prove each axiom or your theorem doesn't hold together. The CAGW claim is that warming exists AND is human caused AND is

That is how science works - you have to prove each axiom or your theorem doesn't hold together. The CAGW claim is that warming exists AND is human caused AND is going to continue AND is something we can do something about AND the cost of action is less than dealing with the consequences of inaction. A skeptic could quite reasonably have been skeptical about any or all of those assertions at various points in time. If you manage to establish some of them to a satisfactory degree, that doesn't give you a free pass to assume away all the others.