Climbing stairs is different. How do I know this?
For the last week I have been riding my bike fitted with a pair of "Power Cranks" (www.powercranks.com, no comercial conection with me). These are cranks that have their own one way clutch (freewheel) built in to them, making the cranks independat of each other. If you want to ride you have to lift your feet as well as pushing them down. The idea is to make you pedal in super efficient neat circles, a pedal training devise. For me the jury is still out on that one and will be for some time.
When you climb stairs, you have to lift your feet up as well as pushing down. When you ride a bike, you do not have to lift your feet, as pushing down lifts them for you. After 25 years of cycling, I always though I was quite good at lifting my feet. I now know better!