First-Order Logic with Natural Deduction (constructive and classical
versions). For a classical sequent calculus, see Isabelle/LK.
Useful references on First-Order Logic:
Simon Thompson, Type Theory and Functional Programming (Addison-Wesley,
1991) (The first chapter is an excellent introduction to natural deduction
in general.)
Antony Galton, Logic for Information Technology (Wiley, 1990)
Michael Dummett, Elements of Intuitionism (Oxford, 1977)
Examples for First-Order Logic.