Proof:

Statement (i) follows from the uniqueness of lifts. Statement (ii) follows immediately from the definition. To prove (iii) apply the lifting criterion (necessity) to both $\phi$ and $\phi^{-1}$. To prove (iv) apply lifting criterion (sufficiency) to get continuous functions $\phi : \widetilde X \longrightarrow \widetilde X$ and $\psi : \widetilde X \longrightarrow \widetilde X$ such that

$$p\circ\phi = p,\; \phi(\tilde x_1) = \tilde x_2;\quad p\circ\psi = p,\; \psi(\tilde x_2) = \tilde x_1.$$

Then $\phi\circ \psi$ and $\psi\circ\phi$ are both lifts of the map $p : \widetilde X \longrightarrow X$ such that

$$\phi\circ \psi(\tilde x_2) = \tilde x_2,\quad \psi\circ\phi(\tilde x_1) = \tilde x_1$$

The identity map on $\widetilde X$ is also a lift of $p$ with these initial conditions. By uniqueness, we see that both $\phi\circ \psi$ and $\psi\circ\phi$ must be the identity map on $\widetilde X$ proving that $\phi$ and $\psi$ are homeomorphisms. The uniqueness clause follows from the uniqueness of lifts. $\square$