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We propose a novel framework for Alzheimer's disease (AD) detection using brain MRIs. The framework starts with a data augmentation method called Brain-Aware Replacements (BAR), which leverages a standard brain parcellation to replace medically-relevant 3D brain regions in an anchor MRI from a randomly picked MRI to create synthetic samples. Ground truth "hard" labels are also linearly mixed depending on the replacement ratio in order to create "soft" labels. BAR produces a great variety of realistic-looking synthetic MRIs with higher local variability compared to other mix-based methods, such as CutMix. On top of BAR, we propose using a soft-label-capable supervised contrastive loss, aiming to learn the relative similarity of representations that reflect how mixed are the synthetic MRIs using our soft labels. This way, we do not fully exhaust the entropic capacity of our hard labels, since we only use them to create soft labels and synthetic MRIs through BAR. We show that a model pre-trained using our framework can be further fine-tuned with a cross-entropy loss using the hard labels that were used to create the synthetic samples. We validated the performance of our framework in a binary AD detection task against both from-scratch supervised training and state-of-the-art self-supervised training plus fine-tuning approaches. Then we evaluated BAR's individual performance compared to another mix-based method CutMix by integrating it within our framework. We show that our framework yields superior results in both precision and recall for the AD detection task.