The phrase "with this proviso" is often used to introduce a condition or a limitation to a statement. Some synonyms for this phrase include "under these circumstances," "provided that," "on the condition that," "subject to," "as long as," "if and only if," and "with the understanding that." These terms signify that something needs to happen in order for a situation to apply. They highlight the fact that a particular offer, opportunity, or statement is contingent on certain guidelines being followed. These phrases serve as a reminder that things are not always straightforward and that success is often dependent on adhering to stipulations.