No, in general, a true $\Delta^0_1$ assertion may not necessarily be provably $\Delta^0_1$ in a given theory. For example, assume $\text{Con}(\text{PA})$ is true, and consider the formula $\phi(a)$ asserting that $a=a$ and the formula $\psi(a)$ asserting that "$a$ is not the code of a proof of a contradiction in $\text{PA}$," which is expressible as saying that $a$ does not solve a certain specific diophantine equation.
Since we assumed there is no such proof, we have that $\exists a\ \psi(a)$ is equivalent to $\forall a\ \psi(a)$, since these are both true sentences. But there can be no proof of this equivalence in $\text{PA}$, if it is consistent, since $\text{PA}$ proves the former sentence, but if it were to prove the latter sentence, it would be proving its own consistency.
