Opoitsev states in 'A Converse to the Principle of contracting Maps', 1976:
Equivalent metrics are also topologically equivalent. The converse is false, but the following easily checked statement is true: If two metrics d1 and d2 are topologically equivalent and the spaces (X, d1) and (X, d2) are both complete, then d1 and d2 are equivalent.
He says it is easy to check, but I am not sure. Is that statement true and if so, how can it be proved? Or are there other necessary conditions for topological equivalence to imply equivalence?