Breastfeeding has shown small signs of boosting the immune system but it is debatable. 

Almost every other claim about the benefits of breastfeeding have been shown to be null. Many studies fail to recognize correlation between women who breastfeed and other factors such as age, class, IQ, etc. 

The problem with studies about mothers and babies is that they are extremely rare and difficult to justify. What few studies we have are often quite misleading. But if you eliminate poorly done studies you see a clear picture that formula is really perfectly fine for a babies health. In a strange twist, the strongest evidence against formula is that breastfeeding can slightly prevent breast cancer in the mother.