Definitely, Bitcoin usage will reduce the demand for fiat and increase the demand for bitcoins which in turn increase the price of bitcoins.That is why many governments fear bitcoins.
The fact that most of bitcoiners will end up converting their bitcoin to fiat means the demands for the fiat itself still remain the same and there's nothing for the government to be afraid of about bitcoin.
And as you know that bitcoin's value always determined based on fiat currency such as USD and that indirectly reflects that bitcoin still need fiat