We investigate a first-order conditional probability logic with equ-ality, which is, up to our knowledge, the first treatise of such logic. The logic, denoted $\operatorname{LFPOIC}^{=}$, allows making statements such as: $CP_{\geq s}(\phi,\theta)$, and $CP_{\leq s}(\phi,\theta)$, with the intended meaning that the conditional probability of $\phi$ given $\theta$ is at least (at most) $s$. The corresponding syntax, semantic, and axiomatic system are introduced, and Extended completeness theorem is proven.