Machine learning support for medical decision making is truly helpful only when it meets two conditions: high prediction accuracy and a good explanation of how the diagnosis was reached. Support vector machines (SVMs) successfully achieve the first target due to a kernel-based engine; evolutionary algorithms (EAs) can greatly accomplish the second owing to their adaptable nature. In this context, the current paper puts forward a two-step hybridized methodology, where learning is accurately performed by the SVMs and a comprehensible emulation of the resulting decision model is generated by EAs in the form of propositional rules, while referring only those indicators that highly influence the class separation. An individual highlighting of the medical attributes that trigger a specific diagnosis for a current patient record is additionally obtained; this feature thus increases the confidence of the physician in the resulting automated diagnosis. Without loss of generality, we aim to model three breast cancer instances, for reasons of both high incidence of the disease and the large application of state of the art artificial intelligence methods for this medical task. As such, the prediction of a benign/malignant condition as well as the recurrence/nonrecurrence of a cancer event are studied on the Wisconsin corresponding data sets from the UCI Machine Learning Repository. The proposed hybridization reached its goals. Rule prototypes evolve against a SVM consistent training data, while diversity among the different classes is implicitly preserved. Feature selection eventually leads to a resulting rule set where only the significant medical indicators together with the discriminating threshold values are referred, while individual relevance of attributes can be additionally obtained for each patient. The gain is thus dual: the EA benefits from a noise-free SVM preprocessed data and the resulting SVM model is able to output rules in a comprehensible, concise format for the physician.