> Have there actually been studies that prove Uber is significantly cheaper than cab?

Where I'm from (Toronto) the taxi rates are fixed by law and the Uber rates are publicly available on their website.

Taxi rates: $3.25 + $1.74 per km + $0.50 per minute

UberX rates: $2.50 + $0.80 per km + $0.18 per minute

It's not even close.