Ok, I see some confusion here.
Ы and И have nothing to do with vowel reduction. They are different vowels, and so they affect previous consonants in different ways. Or rather Ы does not affect them at all, and И palatalize them.
It means that most consonants in Russian can be EITHER soft OR hard, depending on the following vowel.
In your example the first "б" is soft (palatalized by following И) - and is pronounced as such, and the second "б" is hard.
Vowels that palatalize previous consonants: ё, и, ю, е, я.
Their alter egos that DO NOT palatalize previous consonants: ы, у, э, а.
For example,
люк (hatch) - лук (onion)