Resources for learning formal math?
For an undergraduate-level book, none of these three can do you wrong:
- Mendelson, Introduction to mathematical logic
- Enderton, A mathematical introduction to logic
- Boolos, Burgess, and Jeffrey, Computability and logic
They certainly go into the details, and they will leave you in a position where you could go further. If you can only look at one, try Mendelson.
An older text, which is now reprinted by Dover, is Kleene's Mathematical logic. It is also very thorough, and has the advantage of being inexpensive. (Be aware that Kleene's Introduction to metamathematics is a completely different book, which is much more advanced.)
There is a good list on this website. I can personally recommend the Walicki notes (free). It includes a good introduction to modal logic, so that's always nice.
Of the books, Geoffrey Hunter's Metalogic is very good and covers all bases.
If you're still looking for more resources here, you might want to look at
- The Schuam's Outline of Logic which uses a Fitch-Jaskowski style natural deduction calculus (though it doesn't call it such), though admittedly the rule of substitution doesn't get used much here.
The next four axiomatically develop classical logic:
- Jan Lukasiewicz's Elements of Mathematical Logic.
- A. N. Prior's Formal Logic.
- The Metamath site.
I'd think the Metamath site closest to what you seek.