"On condition" is a phrase that refers to something that is dependent on certain terms or requirements being met. There are various synonyms for this phrase that can be used to convey the same meaning, such as "provided that", "assuming that", "if", "as long as", "provided", "presuming", "subject to", "contingent on", "given that", "relying on", "hinging on", and "based on". Each of these synonyms carries a slightly different nuance, which can be used to reflect the specific context and tone of the text. Regardless of which synonym is used, the underlying meaning remains the same: something is conditional and depends on certain stipulations being met.