&gt;You mean like seeding the system against diseases the child may not encounter otherwise? It doesn't just speed up, it literally seeds the system and helps the mothers immune system step in and boost the babies. 

Except they get those same antibodies from the saliva of their mother in almost all cases. 

&gt;babies that had been breastfed for at least 6 months 

No, babies who drank breast milk show reduced signs of allergies (though rarely is it significant). Not breastfeeding. The two are not the same thing. A mother who doesn't produce milk and gets some from elsewhere for the immune system boosting benefits, but mostly feeds their child formula, gets the exact same advantages. A woman who only produces enough for 1/8th the babies diet and supplements it with formula is still getting the same advantages.