Bitcoin will always be dependent on fiat. Only if people were to completely accept bitcoin would fiat be rendered useless.
Yeah, and I don't see it happening in near future as the market cap of fiat is huge as compared bitcoin, so bitcoin would be surely dependent on fiat in future too.
There are chances if everything goes digital then we can expect that people would use more bitcoins as a payment method in compare to fiat, but that doesn't mean fiat will simply disappear.