This hadith has been transmitted on the authority of Uthman and A'isha with the same wording.
Sahih Muslim Book 44, Hadith 41
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Then I sought permission from him and he got up and raid to A'isha: Wrap yourself well with your cloth, then I got my need fulfilled and came back. And A'isha said: Allah's Messenger, why is it that I did not see you feeling any anxiety in case of dressing properly in the presence of Abu Bakr and 'Umar (Allah be pleased with them) as you showed in case of 'Uthman. Thereupon Allah's Messenger (ﷺ) said: Verily Uthman is a person who is very modest and I was afraid that if I permitted him to enter in this very state he would not inform me of his need.
Sahih Muslim Book 44, Hadith 40 Next Hadith → Open it for him and give him glad tidings of Paradise and, lo, it was Abu Bakr. I opened (the gate) for him and gave him the glad tidings of Paradise. Then another person asked for the door to be opened, whereupon he said: Open it and give him the glad tidings of Piradise. He said: I went away and, lo, it was 'Umar. I opened it for him and gave him the glad tidings of Paradise. Then still another man asked for the door to be opened, and thereupon Allah's Apostle (ﷺ) said: Open it and give him the glad tidings of Paradise after a trial would afflict him. I went and, lo, it was 'Uthman b. 'Affan. 1 opened the door and gave him the glad tidings of Paradise and informed him (what the Prophet had said). Thereupon he said: O Allah, grant me steadfastness. Allah is one Whose help is to be sought.
Sahih Muslim Book 44, Hadith 42
Sahih Muslim Book 44, Hadith 40 Next Hadith → Open it for him and give him glad tidings of Paradise and, lo, it was Abu Bakr. I opened (the gate) for him and gave him the glad tidings of Paradise. Then another person asked for the door to be opened, whereupon he said: Open it and give him the glad tidings of Piradise. He said: I went away and, lo, it was 'Umar. I opened it for him and gave him the glad tidings of Paradise. Then still another man asked for the door to be opened, and thereupon Allah's Apostle (ﷺ) said: Open it and give him the glad tidings of Paradise after a trial would afflict him. I went and, lo, it was 'Uthman b. 'Affan. 1 opened the door and gave him the glad tidings of Paradise and informed him (what the Prophet had said). Thereupon he said: O Allah, grant me steadfastness. Allah is one Whose help is to be sought.
Sahih Muslim Book 44, Hadith 42