A very good book along the lines you seem to want is Gallavotti's Statistical Mechanics - A Short Treatise, which can be downloaded from here. He covers many of the classical topics, with a detailed discussion of foundational issues, the role of ergodicity/mixing, etc.
From a very different point of view, with a colleague, we are writing a mathematically rigorous introductory book on the equilibrium statistical mechanics of lattice systems. We hope to finish it for Summer 2016, but you can already download drafts of several chapters here. We are trying hard to make the book as readable and student-friendly as possible.
Simon's book The Statistical Mechanics of Lattice Gases is somehow similar to our book, but covers less material (but more in depth; a second volume was planned, but will most likely never exist). It also covers quantum models.
Another book similar to ours, but much harder, is Georgii's famous Gibbs Measures and Phase Transitions. (In a sense, our book can be considered as an introduction to this book, although we do/will cover some important topics that are not discussed the latter.)
There are of course other books aimed at mathematicians/mathematical physicists on more specific topics (e.g., disordered systems, integrable models, relations with large deviations theory or dynamical systems, etc.).