Deriving $\Phi\vdash\Delta$ from $\lnot\lnot\Phi\vdash\Delta$ without cut rule?

Question

I revisited the old post of mine and I am confused with the answer. The cut elimination theorem states that for any sequent that is derived with cut rule, there exists a derivation of the same sequent without using cut rule. It seems like that the answer suggested that it is impossible to derive $\Phi\vdash\Delta$ from $\lnot\lnot\Phi\vdash\Delta$ without cut? Is it a counterexample to cut elimination theorem? If not, how do I derive $\Phi\vdash\Delta$ from $\lnot\lnot\Phi\vdash\Delta$ without cut?