"提供证据之责任" can be translated into English as "the burden of producing evidence" or, in a more formal or legal context, "the onus of adducing evidence".
"The burden of producing evidence" is a commonly used phrase in legal and everyday contexts, referring to the obligation of a party to present evidence in support of their case or claim.
"The onus of adducing evidence" is a more formal and legalistic expression, where "onus" means the burden or responsibility, and "adducing evidence" means presenting or introducing evidence in court or a legal proceeding.