Re-encrypting a message and proving that the message has not changed
I  think this is not compatible with semantic security (IND-CPA). If such a proof exists,  an adversary that chooses $(m_0,m_1)$ and receives $c_b=E_1(m_b)$ may compute $E_2(m_0)$ and try to prove that $m_b=m_0$. If the proof is correct, then he guesses that $b=0$, else $b=1$. But may be there exist weaker security models that allow to test the equality of two plaintexts.
Your problem seems similar to plaintext checkable encryptions.