Quantifiers

I have been thinking about quantifiers, and the notation around them...since most books use differing notation that is not easy to read.

Since I am trying to be consistent throughout all of mathematics, it seems natural to suggest that the colon ":" should be read as "such that".

In this case, one should write "∃x : P(x)" since one usually writes in natural language "There is some x such that P(x)".

Likewise we often find expressions "If x∈X, then P(x)"...so it would be natural to use the notation "∀x∈X, P(x)" or "∀ x, x∈X and P(x)". After all, "x∈ X" is really a predicate "isIn(x,X)"...

I'm currently revising my notes on logic in Fascicles 0 of my Elements of Mathematics, which is why I bring this notational problem up! Hopefully, I will be done with revising and improving my logic chapter soon.