The questioning is not whether it will but when.
The dollar will continue to exist, but it will be as it is today. The dollar will be replaced by bitcoin as the currency of value. This does not depend on governments, institutions and anything. It only depends on whether or not people accept bitcoin as the bargaining chip. What is the dollar's intrinsic value? What is Bitcoin's intrinsic value? This answer is completed by what people will accept and value these two monetary agents.
No government will be able to control bitcoin, because it does not depend on it to create more Bitcoin, different from the dollar.
When the global economy concludes that the US is unreliable to pay its debts, and this is close to happening, bitcoin will be the viable alternative to replace it. Whether you want it or not. This cryptocoin will serve as ballast, and there will be purchases and sales, transfers of values ​​between countries based on dollars but in Bitcoin.

Note: When it comes to bitcoin, it may not be it, but any cryptocoin that fulfills the requirements of a good coin according to the concepts of Aristotle.