Ice Cores

Ice cores are used to demonstrate on charts what the global mean temperature was thousands of years ago.  I read this wikipedia article on Ice Core and it was talking about how the cores only show local temperature.  Put this great bit of knowledge together with today's explanation of cooler temperatures being a side-effect of global warming because an increase in global temperatures will cause a greater disparity of local temperatures, up or down.  So, how can we trust the temperatures in just a few locations to show us to any degree of accuracy of what the temperature really is?