Beyond the foundational significance, the problem of bounding nonlocal correlations by reasonable physical principles has meaningful practical consequences, particularly for device-independent (DI) cryptographic security. In this work, we advance in this direction, demonstrating that the IC is enough to ensure DI security on quantum key distribution (QKD) protocols. Security is proven for a range of theoretically quantum-attainable parameters against individual attacks by a potentially post-quantum eavesdropper. This result follows as a consequence of a strong form of monogamy of Bell's inequality violations, which we have proven to be implied by the recently proposed multipartite formulation for IC. Additionally, we demonstrated that the original bipartite formulation of IC fails to imply monogamy relations and hence, ensure security of DIQKD, thus stressing the necessity of the multipartite framework.