She was so angry that she tore up the letter.
Can I rewrite the sentence like the following?
She was too angry not to tear up the letter.
In English, I'd like to know whether 'not to-infinitive' can be used after 'too + adj'.
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Yes, you can rewrite the sentence that way. It is a known structure, although it is not as common as it used to be and sounds a little old-fashioned and formal.
There is a very slight difference in meaning, I think - the too adjective not to verb form has, to me, an implication of "so adjective that it was impossible or unimaginable for her not to verb". Consider "The chess champion was so smart she saw the trap her opponent was laying" vs. "the chess champion was too smart not to see the trap her opponent was laying." The second form, I think, implies that of course she was going to see it, whereas the first form is closer to just stating that seeing the trap was very smart.