Fiat isn't going to cease to exist. Bitcoin will continue to take 'market share' from fiat currencies though and as its price becomes more stable this speed will only increase. The fact is though, is that Bitcoin does not function (at least at this stage) like a currency should do.

No-one in their right mind would spend their bitcoin now, if they truly believe it will significantly increase in value in the long term, which if you are investing in Bitcoin, you must therefore believe. Modern capitalist economies are built on consumption, which requires consumers to spend money. Bitcoin does not incentivise this at all. The inflation and low interest rates of Fiat currencies do. For example I own Bitcoin and was out in town the other night with some friends for a few drinks. Low and behold I find out that the bar accepts Bitcoin. However I did not choose to pay in Bitcoin because I believe the bitcoin I would spend today, will be worth a lot more in the future. Therefore as a currency this does not work. It does however fit the profile of a digital Gold and a means to store value.

In my eyes we will see cryptocurrencies acting as an alternative to fiat but they will never replace them.