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.)