Introduction-to-proof book using natural-deduction-style rules?

Can anyone recommend a textbook for an introduction-to-proofs bridge course that discusses the rules for "proving and using" (aka "introduction and elimination") each connective and quantifier, as in type theory or natural deduction? All the books I've looked at so far either explain the connectives using truth tables, or don't explain them specifically at all. They often discuss some of the rules (like "direct proof" = implies-intro and "proof by cases" = or-intro and "constructive proof" = exists-intro), but usually just in a list of "proof methods" without an organizing structure.

(To clarify, I'm not looking for a textbook in formal logic. I know that some people use textbooks in formal logic for bridge courses, but I don't think that would be appropriate for my students. I want a textbook which introduces students to the idea of proof, to basic concepts in mathematics like induction, divisibility, and sets, and to other aspects of mathematics like mathematical writing and exploration / proof search. There are lots of such textbooks, but I haven't found one yet that organizes the proof rules by their governing connectives as above.)

• Have you looked at How to Prove It by Daniel Velleman? I bought the book my senior year of high-school and I believe it introduced me to advanced math pretty well. An entire chapter is devoted to proof strategies and different kinds of proofs. Nov 23 '14 at 4:42
• @AndreyKaipov no, I hadn't yet, thanks! His chapter 3 is pretty much exactly what I wanted. If you post that as an answer, I'll accept it. Nov 25 '14 at 23:34
• I've written a text that addresses your needs, I believe; however, it's not available (yet) publicly. How can I share it with you (and others who may be interested)? Dec 2 '14 at 5:18
• @brendansullivan07, if you only want to share it with me, my email address is on my web site; you can send me an attachment or point me to a private URL, dropbox folder, etc. I can't think of any way to share it with others who may be interested yet keep it non-public other than inviting any such person to contact you directly. Dec 2 '14 at 7:21
• @StevenGubkin I wouldn't really use the phrase "weasel out", but I won't object to it either. I would say that $p\to \bot$ is really just the definition of $\neg p$. My paper about linear logic involves notions of "refutation" that are dual to "justification" when discussing semantics, but the rules of proof don't involve separate notions of "proving false" and "proving true". Feb 22 at 0:23