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I wondered about some good books that aren't necessarily "introductory", but delve deeper into the "art of proof-writing".

  • I believe the art comes from the writing aspect; so consider books on writing. Although different proofs of the same theorem is artistic itself. – Alberto Takase Feb 04 '19 at 21:01
  • You are re-asking a question that has many good answers already. Please don’t do that: do a search before you post. You will find many more than the duplicate I suggested that suit your question. – rschwieb Feb 05 '19 at 04:14

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On my side I enjoyed Mathematical Writing from Franco Vivaldi. English is not my mother language and the book was very useful.

mathcounterexamples.net
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I appreciated reading this one, very well written, but at an introductory level. Being simple allows the reader to focus on the main purpose of the book, which is how to efficiently combine all available information to proof or reject a proposition:
Daniel J. Velleman, 2006, How to Prove it: A Structured Approach: Second edition, Cambridge University Press 978-0-521-86124-3.

EDIT: I just saw that your question is related to this one: Book for proof writing

Bertrand
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