Breast cancer, which attacks the glandular epithelium of the breast, is the second most common kind of cancer in women after lung cancer, and it affects a significant number of people worldwide. Based on the advantages of Residual Convolutional Network and the Transformer Encoder with Multiple Layer Perceptron (MLP), this study proposes a novel hybrid deep learning Computer-Aided Diagnosis (CAD) system for breast lesions. While the backbone residual deep learning network is employed to create the deep features, the transformer is utilized to classify breast cancer according to the self-attention mechanism. The proposed CAD system has the capability to recognize breast cancer in two scenarios: (Binary classification) and (Multi-classification). Data collection and preprocessing, patch image creation and splitting, and artificial intelligence-based breast lesion identification are all components of the execution framework that are applied consistently across both cases. The effectiveness of the proposed AI model is compared against three separate deep learning models: a custom CNN, the VGG16, and the ResNet50. Two datasets, CBIS-DDSM and DDSM, are utilized to construct and test the proposed CAD system. Five-fold cross validation of the test data is used to evaluate the accuracy of the performance results. The suggested hybrid CAD system achieves encouraging evaluation results, with overall accuracies of 100% and 95.80% for binary and multiclass prediction challenges, respectively. The experimental results reveal that the proposed hybrid AI model could identify benign and malignant breast tissues significantly, which is important for radiologists to recommend further investigation of abnormal mammograms and provide the optimal treatment plan.