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Is it possible to automatically export all theorems to single files, say PNG or other image formats?

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There is a way to automatically export all theorems to a single new document. See https://tex.stackexchange.com/a/46179/154418

After that, you could use a \AtEndEnvironment{theorem}{\pagebreak}, which should yield you one theorem per page.

Now, for the image part, using the convert option of the standalone class will directly output to PNG, with transparency:

\documentclass[convert]{standalone}

If you want one image per page, it might be simpler to chain it with a PDF to multiple PNG external tool, depending on what's available on your system.

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