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First chapter of Jean-Yves Girard, "Proof Theory and Logical Complexity", Vol I, Bibliopolis, 1987

It satisfy satisfies all of your conditions, but it is not an elementary book. If I remember correctly, the authors (A.S. Troelstra and H. Schwichtenberg) of the book "Basic Proof Theory" which is published in 2001 wrote in their introduction that their intention was to fill the gap between this and all other (introductionary) books in proof theory. As far as I know, he never published the second volume.
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First chapter of Jean-Yves Girard, "Proof Theory and Logical Complexity", Vol I, Bibliopolis, 1987

It satisfy all of your conditions, but it is not an elementary book. If I remember correctly, the authors (A.S. Troelstra and H. Schwichtenberg) of the book "Basic Proof Theory" which is published in 2001 wrote in their introduction that their intention was to fill the gap between this and all other (introductionary) books in proof theory. As far as I know, he never published the second volume.