〜ずにはおかない・〜ないではおかない
is bound to (make someone feel); will definitely
Verb ない-stem + ずにはおかない (する → せずにはおかない)
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ずにはおかない has two uses.
With a causative verb, something is bound to affect people: 彼の演技は、観客を感動させずにはおかない, "his performance can't fail to move the audience". The subject is a work, event or words, and the effect is on others' feelings: 感動させる, 泣かせる, 考えさせる, 心を動かす.
With a person as the subject, it's a strong resolve: 今度こそ、真相を明らかにせずにはおかない, "this time, I'll uncover the truth whatever it takes".
It's literally "won't leave it without doing". Compare ずにはいられない (N2), which is about your own irresistible urge: 笑わずにはいられない, "I can't help laughing".
Examples
彼の演技は、観客を感動させずにはおかない。
His performance can't fail to move the audience.
この映画は、見る人を泣かせずにはおかない。
This film is bound to make anyone who sees it cry.
今度こそ、真相を明らかにせずにはおかない。
This time, I'll uncover the truth whatever it takes.
彼の言葉は、人の心を動かさずにはおかない。
His words can't fail to move people's hearts.
警察は、必ず犯人を捕まえずにはおかないだろう。
The police won't rest until they've caught the culprit.