Im having a problem with the first rule for class usage enunciated in a book on set theory. ("Basic Set Theory" by Azriel Levy)
T Its the one where a predicate infers some other predicate. It is not clear to me whether A is a set, a class element or a set element and the same for x. The phi's are predicates. I think what is intended to say is that, if a predicate is valid for some element in a set, then a classe can be constructed by all objects satisfying that same predicate Thank you in advance.
From pg 11 of the book: "we shall use upper case Roman letters for class variables". Likewise, set variables are lower-case.
The meaning of the inference rule is summarized right under 4.2 as: "whatever holds for all classes holds for all sets." $\Phi$ is just some formula, and this has nothing to do with comprehensions. We are interpreting a formula as a universal statement. $\Phi(A)$ means "$\Phi(A)$ holds for all classes $A$" and from that we can infer $\Phi(x),$ which means "$\Phi(x)$ holds for all sets $x$".