Let $\alpha_1$; $\alpha_2$ be any two different finite binary strings. Let $E_{\alpha_1\alpha_2}$ be the set of all codes of programs M such that M does not distinguish between the input $\alpha_1$ and the input $\alpha_2$. That is, if M halts on one of these inputs, it halts on the other as well, and in that case, M($\alpha_1$) = M($\alpha_2$). Prove that the decision problem defined by $E_{\alpha_1\alpha_2}$ is not decidable.
I think I can prove that it's undecidable using the halting problem but I am not sure how to use it. What is a general method to solve these types?