Local transformations of superpositions of entangled states

Suppose that we have two entangled states |ϕ1〉, |ψ1〉 that cannot be converted to any of other two states |ϕ2〉, |ψ2〉 by local operations and classical communication. We analyze the possibility of locally transforming a superposition of |ϕ1〉 and |ψ1〉 into a superposition of |ϕ2〉 and |ψ2〉. By using the Nielsen's theorem we find the necessary and sufficient conditions for this conversion to be performed.