Visibility of a planet on the other side of the sun at midnight

Question

Context: I've been having a discussion with my vectors and mechanics professor (course Mathematics BSc) about a problem on a recent coursework. The following is the model I came up with.

Define:

1. $S$ the Sun;
2. $E$ the Earth;
3. $P$ some other planet in the solar system;
4. $O$ the position of an observer somewhere on $E$;
5. $0<l<90$ the latitude of $O$;
6. $\Pi_E$ the orbital plane of $E$, the $xy$ plane;
7. $\Pi_P$ the orbital plane of $P$;
8. $\Pi_O$ the plane tangent to $E$ at $O$;
9. $\alpha$ the acute angle between $\Pi_E$ and $\Pi_P$;
10. $\beta=90-l$ the acute angle between $\Pi_E$ and $\Pi_O$.

Assume:

1. The local time at $O$ is midnight;
2. The date is the Summer solstice;
3. The visual sizes of $P$ and $S$ are both non-zero but otherwise negligible;
4. The projections of $E$, $S$ and $P$ onto the $xy$ plane are roughly but not exactly collinear;
5. The solar system is Euclidean;
6. Light travels in straight lines without attenuating;
7. Relative to $E$, $P$ is on the far side of the line of intersection between $\Pi_P$ and $\Pi_O$.

Given these assumptions, I think that $P$ is visible from $O$ if and only if $\alpha>\beta$. Am I right?

Edit: On reflection, I think assumption 7 is the true condition for visibility.