Lower semicontinuous multivalued mappings \(\Phi\), \(\Psi : X \to 2^Y\) are considered, where \(X\), \(Y\) are Banach spaces. Under some assumptions (closedness and convexity of the values of \(\Phi \) and \(\Psi \), surjectivity of \(\Phi \), a certain compactness of \(\Psi\), etc.), it is shown that \[ \dim (\{x \in X;\;\Phi (x) \cap \Psi (x) \neq \emptyset \}) \geq \dim (\Phi ^{-1}(0)), \] where \(\dim \) denotes the covering dimension. Two applications to differential inclusions are given.