What sort of gear do you need for conditions like that? During one cold snap, my chain literally froze on a long ride through some light rain/sleet. (No, I don't know how that happened, but it did.)
And that was in the Seattle area, where it rarely gets all that cold.
I was a Mormon missionary in Hokkaido in some of the snowiest places in the world and got around just fine using a cheap mountain bike. The only additional "equipment" needed was a lighter, to thaw out my bike lock when it would freeze shut.
And that was in the Seattle area, where it rarely gets all that cold.