Conditional Feature Importance (CFI) is a classical variable importance measure that accounts for the relationship between the studied feature and the others. However, CFI has not yet been studied from a theoretical perspective because the conditional sampling step has generally been considered a purely practical problem. In this article, we demonstrate that the recent Conditional Permutation Importance (CPI) is indeed a valid implementation of this concept. Under the conditional null hypothesis, we then establish a double robustness property that can be used for variable selection. With either a valid model or a valid conditional sampler, the method correctly identifies null coordinates. Under the alternative hypothesis, we study the theoretical target and link it to the popular Total Sobol Index (TSI). We introduce the Sobol-CPI, which generalizes CPI/CFI, prove that it is nonparametrically efficient, and provide a bias correction. Finally, we propose a consistent and valid type-I error test and present numerical experiments that illustrate our findings.