I have some Algebraic Topology notes that prove the following Lemma:
If $X$ is simply connected and locally path connected, then every covering projection $p:Y \to X$ is trivial (i.e. the whole space $X$ is evenly covered by $p$).
I understand this, and I understand the proof (it constructs a homeomorphism $Y \cong X \times D$ where $D$ is a discrete space). The proof is followed by the comment: "It follows that if a locally path connected space has a simply connected covering space, then the latter is unique up to homeomorphism".
I don't understand why this is true. I can see it's true if we transpose "locally path connected" and "simply connected", since $X$ is lpc iff $Y$ is. Otherwise, I don't really have a clue.
I'd appreciate any enlightenment. Thanks!