Perfect secrecy theorem Let $E = (E,D)$ be a Shannon cipher defined over $(K,M, C)$. Consider a
probabilistic experiment in which $k$ is a random variable uniformly distributed over $K$. Then $E$ is perfectly secure if and only if for
every predicate $Φ$ on $C$, for all $m_0,m_1 2M$, we have

$Pr[Φ(E(k,m_0))] = Pr[Φ(E(k,m_1))]$

I dont understand the meaning of $Φ$ in this theorem
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 &amp; 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.