Not sure I understand what's happening, but three suggestions:

1. [math] (A = B) \Leftrightarrow (A \subset B \land B \subset A)[/math]
2. [math] (A = B) \Leftrightarrow \forall a (a \in A \leftrightarrow a \in B)[/math]
3. [math] (A = B) \Leftrightarrow \forall P, (A \in P \leftrightarrow B \in P)[/math]

(1) is the definition of the identity relation for sets. (2) is an extensionality principle for identity. (3) is the principle of the identity of indiscernibles.