Derivation of the Principle of Extensionality from Basic Law V

[Note: We use $$\epsilon F$$ to denote the extension of the concept $$F$$.]

Assume Extension(x) and Extension(y). Then $$\exists F(x\eqclose \epsilon F)$$ and $$\exists G(y\eqclose \epsilon G)$$. Let $$P,Q$$ be arbitrary such concepts; i.e., suppose $$x\eqclose \epsilon P$$ and $$y \eqclose \epsilon Q$$.

Now to complete the proof, assume $$\forall z(z\in x \equiv z\in y)$$. It then follows that $$\forall z(z\in\epsilon P\equiv z\in\epsilon Q)$$. So, by the Law of Extensions and the principles of predicate logic, we may convert both conditions in the universalized biconditional to establish that $$\forall z(Pz\equiv Qz)$$. So by Basic Law V, $$\epsilon P =\epsilon Q$$. So $$x = y$$.