Fuzzy gradual rules of the form the more X is A, the more Y is B linguistically express information about the correlation between attributes and their co-variation. They thus provide valuable information summarizing the trends observed in a given data set. In this paper, we consider strengthened fuzzy gradual rules, i.e. gradual rules enriched with a clause introduced by the expression "all the more": such rules of the form the more X is A, the more Y is B, all the more Z is C offer additional precisions on the relation between the attributes. We study the definition of such strengthened rules, discussing their possible semantics, considering several interpretations of fuzzy gradual rules. We then propose quality criteria as well as a mining algorithm.