If this is a noob question, apologies, I am a noob.
So, I understand the understand the colon in the sense of (Book of Proof; Hammack; 2018):
$$X=\{epression:rule\}$$
And certainly I see the use of colon in something like the following to give a name that stands for an expression/rule:
\begin{matrix} [\alpha..\beta] & \text{closed interval}: \text{the set }\{x \in \mathbb{R}\mid \alpha \leqslant x \leqslant \beta \} \end{matrix}
Remembering we are talking usage of ":", so starting with this formula (sans ":"):
$$\forall *.\phi_{1}\land ... \land \phi_{n} \to \varphi$$
I found this formula in a paper that references out to Antony Galton (Logic for Information Technology, John Wiley & Sons, Chichester, England, October 1990). That formula, from Galton according to the referenced paper, is an "implication formula" denoted by $\psi$. Since the point of having a symbol to stand for the expression was to use the symbol in other expressions, I opted to summarise that in my paper as:
$$\psi:\forall *.\phi_{1}\land ... \land \phi_{n} \to \varphi$$
My justification was that "Book Of Proof"(Hammack, 2018) opts into explaining use of letters such as P, Q, R and S to stand for specific statements. Galton clearly was using $\psi$ rather than P, Q, R or S to stand for "implication formula". The colon was used in Hammack to separate the symbol from the statements. I take Hammack to be implying approach is common knowledge.
Is there a survey text on math or proof, other than Hammack, that introduces the form:
$$symbol:statements$$