The modal operator $\square$ refers to necessity; its dual, $\lozenge$, refers to possibility. (A sentence is necessarily true iff it isn't possible for it to be false, and vice versa.) $P(\varphi)$ means that $\varphi$ is a positive (in the sense of "good") property; I'll just transcribe it as "$\varphi$ is good". I'll write out the argument colloquially, with the loss of precision that implies. In particular, the words "possible" and "necessary" are vague, and you need to understand modal logic somewhat to follow their precise usage in this argument.
- Axiom $1$: If $\varphi$ is good, and $\varphi$ forces $\psi$ (that is, it's necessarily true that anything with property $\varphi$ has property $\psi$), then $\psi$ is also good.
- Axiom $2$: For every property $\varphi$, exactly one of $\varphi$ and $\neg\varphi$ is good. (If $\neg\varphi$ is good, we may as well say that $\varphi$ is bad.)
- Theorem $1$ (Good Things Happen): If $\varphi$ is good, then it's possible that something exists with property $\varphi$.
Proof of Theorem $1$: Suppose $\varphi$ were good, but necessarily nothing had property $\varphi$. Then property $\varphi$ would, vacuously, force every other property; in particular $\varphi$ would force $\neg\varphi$. By Axiom $1$, this would mean that $\neg\varphi$ was also good; but this would then contradict Axiom $2$.
- Definition $1$: We call a thing godlike when it has every good property.
- Axiom $3$: Being godlike is good.
- Theorem $2$ (No Atheism): It's possible that something godlike exists.
Proof of Theorem $2$: This follows directly from Theorem $1$ applied to Axiom $3$.
- Definition $2$: We call property $\varphi$ the essence of a thing $x$ when (1) $x$ has property $\varphi$, and (2) property $\varphi$ forces every property of $x$.
- Axiom $4$: If $\varphi$ is good, then $\varphi$ is necessarily good.
- Theorem $3$ (God Has No Hair): If a thing is godlike, then being godlike is its essence.
Proof of Theorem $3$: First note that if $x$ is godlike, it has all good properties (by definition) and no bad properties (by Axiom $2$). So any property that a godlike thing has is good, and is therefore necessarily good (by Axiom $4$), and is therefore necessarily possessed by anything godlike.
- Definition $3$: We call a thing indispensable when something with its essence (if it has an essence) must exist.
- Axiom $5$: Being indispensable is good.
- Theorem $4$ (Yes, Virginia): Something godlike necessarily exists.
Proof of Theorem $4$: If something is godlike, it has every good property by definition. In particular, it's indispensable, since that's a good property (by Axiom $5$); so by definition something with its essence, which is just "being godlike" (by Theorem $3$), must exist. In other words, if something godlike exists, then it's necessary for something godlike to exist. But by Theorem $2$, it's possible that something godlike exists; so it's possible that it's necessary for something godlike to exist; and so it is, in fact, necessary for something godlike to exist. QED.