The short answer is: Yes.
The longer answer:
Even without a ban, we will be driving, or more likely getting rides in, electric cars.
The Tesla model 3 performance model is now comparable with BMW in price. It is faster, quieter, cheaper and easier to charge, the average driver could commute to and from work for a week between charges. Charging on a long trip will take half an hour every 4 hours (I like to stop every 2 hours or to stretch my legs anyway and have something to eat every 4 - 6 hours) Battery prices are still tumbling so I expect the price to go down as Tesla gets their manufacturing sorted.
That will leave a second hand market full of relatively good but older internal combustion engines, but over time gas stations will not have sufficient customers to continue operating and it will be harder to to find one.
So there will be challenges to overcome but yes inner cities will ban all petrol and diesel cars, because no sane politician would vote against cleaner cities when a cleaner mode of transport is a viable and cost effective option.