This question has the danger of being subjective but I nevertheless think people might learn from each other's answers.
Mathematics textbooks almost never write their proofs in formal logic, but in "mathematical natural language".
What I mean is, that there are two ways to prove a statment:
Using natural language: "take any $\epsilon$. Now define $\delta(\epsilon)$ such that ... Then clearly, for all ..."
Using formal logic with logical deductive rules: $\forall x\in X\exists\delta...$. Applying logic rule $X$ gives $\exists x ...$.
Of course there is a good reason for using natural language: it is more intuitive and therefore easier to follow and easier to come up with.
However I sometimes find that using formal logical rules can clarify my thinking and help me woth proofs. I find that when I get stuck using natural language, switching to thinking about the problem in formal logic can help, and vice versa.
So I'm wondering, do you ever use formal logic while you're working to come up with a proof? Why, why not? What role does it play in your process of coming up with proofs?
Note: I'm only a relative beginner at proving theorems, since I have much more experience with applied math.