That predicate is basically telling you that the resulting ciphertext does not leak any information about the plaintext that has been encrypted.
Assume that there exists some predicate $\Phi'$ such that $Pr[\Phi'(E(k,m_0)] \neq Pr[\Phi'(E(k,m_1)]$, for some $m_0,m_1$ pair. Then, if you are given a ciphertext that corresponds to some of those two messages, computing that predicate you can tell whether $m_0$ or $m_1$ was encrypted, which means that your cipher does not provide perfect secrecy.
For a more precise enunciation, you can check Lemma 2.4 of Katz & Lindell's "Introduction to Modern Cryptography" and its proof, where the predicate used there is $\Phi(c)$ := "$E(k,m)=c$", for $k\in K,m\in M$. And then, you have $Pr[E(k,m_0)=c] = Pr[E(k,m_1)=c]$.
You can try to think of any other predicate. The important thing is that no such predicate must exist.