# Would $\mathsf{P=BPP}$ imply $\mathsf{dIP=IP}$ and if not then why?

Complexity class $$\mathsf{IP}$$ includes all problems that can be solved using an interactive proof system where the verifier is a probabilistic polynomial time machine, and the prover is a machine of an unbounded power. It is known to be equal to $$\mathsf{PSPACE}$$.

Making the verifier a deterministic machine would make a complexity class $$\mathsf{dIP}$$ which is known to be equal to $$\mathsf{NP}$$.

However, if $$\mathsf{P=BPP}$$ would that mean that a deterministic verifier is as powerful as a probabilistic verifier, resulting in some implications believed to be false by the majority of computer scientists?