Grading: Its chain is "sahih".
Technical note / Defect ('illah): Ibn Wahb in the chain refers to "'Abd Allah bin Wahb".

Reference / Source: It was recorded by Imam al-Nasa'i (4751) via Yunus bin 'Abd al-A'la, from Ibn Wahb with this same chain.

Technical note / Defect ('illah): This narration was reported by Nafi', the freed slave of Ibn 'Umar, from Ibn 'Umar (may Allah be pleased with them both), and the narrators differed over its chain. Imam al-Bukhari (may Allah have mercy on him) says in "'Ilal al-Tirmidhi" (455) that (most of) Nafi''s students reported this hadith from Nafi' from Ibn 'Umar as "mawquf" (a statement of the Companion), except Ayyub al-Sakhtiyani, who reports it from Nafi' from the Prophet of Allah (peace and blessings be upon him) as "marfu'". However, the critics say that Ayyub too, in the last period of his life, began reporting it as mawquf.

Reference / Source: This marfu' narration was recorded by Imam Ahmad (4510, 4581, 5093, 5094, 5362, 5363, 6087, 6103, 6104, 6414), Abu Dawud (3261, 3262), Ibn Majah (2105, 2106), al-Tirmidhi (1531), al-Nasa'i (4716, 4752, 4753) and Ibn Hibban (4339, 4342) through routes from Ayyub al-Sakhtiyani, and by Ibn Hibban (4340) and al-Bayhaqi (10/46) through Ayyub bin Musa bin 'Amr al-Umawi; both of them narrate it from Nafi' as "marfu'".

Technical note / Defect ('illah): Imam al-Bayhaqi says that this hadith is known as "marfu'" only from the narration of Ayyub al-Sakhtiyani. This statement suggests that al-Bayhaqi (may Allah have mercy on him) doubted the soundness of the route of Ayyub bin Musa.

Details of the narration: Among the students of Ayyub al-Sakhtiyani, Ma'mar bin Rashid reported it via Nafi' as follows: When Ibn 'Umar (may Allah be pleased with them both) swore an oath he would say, "By Allah, I will not do such-and-such, in sha' Allah"; then he would do it and not pay expiation for the oath (because saying in sha' Allah makes an exception).
Reference / Source: 'Abd al-Razzaq recorded it (16113), and Sufyan al-Thawri also corroborated this "mawquf" narration of Ma'mar, with him at (16115).

Technical note / Defect ('illah): Hammad bin Zayd says that Ayyub al-Sakhtiyani used to report this hadith as "marfu'", then later he abandoned that (out of caution).
Reference / Source: Al-Bayhaqi reported this in (10/46).

Reference / Source: It was recorded by al-Tabarani in "al-Mu'jam al-Awsat" (3075), Tammam al-Razi in "al-Fawa'id" (447) and Abu Nu'aym in "al-Hilyah" (6/79) with the chain Hassan bin 'Atiyyah from Nafi', as marfu'.
Grading: According to al-Tabarani and Abu Nu'aym, 'Amr bin Hashim al-Bayruti is alone in reporting it as marfu', and he is "layyin al-hadith" (a narrator with weak memory).

Technical note / Defect ('illah): 'Ubayd Allah bin 'Umar al-'Umari also narrated it from Nafi', and his students also differed. Abu Mu'awiyah Muhammad bin Khazim narrated it with the chain 'Ubayd Allah from Nafi' as "marfu'", as in Abu al-Shaykh's "Tabaqat al-Muhaddithin" (225) and Abu Nu'aym's "Tarikh Asbahan" (2/140).

Reference / Source: On the other hand, Ibn Jurayj and 'Abd al-Razzaq narrated it from 'Ubayd Allah al-'Umari as "mawquf" ('Abd al-Razzaq 16112).

Reference / Source: Imam Malik narrated it in "al-Muwatta'" (2/477) with the chain Nafi' from Ibn 'Umar, also as "mawquf".

Reference / Source: It was recorded by 'Abd al-Razzaq (16111) from 'Abd Allah bin 'Umar al-'Umari, and by al-Bayhaqi (10/46) through Ibn Wahb, from 'Abd Allah bin 'Umar, Imam Malik and Usamah bin Zayd, all three of them from Nafi' from Ibn 'Umar, as "mawquf".

Grading: Ibn 'Adi (3/86) and al-Bayhaqi (10/47) narrated it with the chain Musa bin 'Uqbah from Nafi' as "mawquf".
Technical note / Defect ('illah): Its chain contains a narrator named Dawud bin 'Ata', who is "da'if".

Details of the narration: Imam al-Tahawi (5/181), al-Daraqutni (4329) and al-Bayhaqi (10/81) narrated through 'Abd al-Rahman bin Abi al-Zinad, from his father, from Salim bin 'Abd Allah, from Ibn 'Umar as "mawquf": "Every exception that is joined (to the oath), there is no breaking of the oath upon the one who says it; and if the exception is not joined (coming after a considerable pause), he is counted as hanith (one who has broken the oath)."

Corroborating chains and supporting reports: The wording given by the author (al-Hakim) is supported by the hadith of Abu Hurayrah (may Allah be pleased with him), found in Musnad Ahmad (13/8088), Ibn Majah (2104), al-Tirmidhi (1532), al-Nasa'i (3855) and Ibn Hibban (1185).
Grading: Its chain is "sahih".

Note / Clarification: Regarding the words of the hadith "lahu thunyahu", 'Allamah al-Sindi writes in his gloss on "Musnad Ahmad": The word "al-thunya" is on the pattern of "al-dunya" and means "exception" (istithna'). That is, this exception benefits him and he does not incur the sin of breaking the oath, whether he does the thing he swore about or not. And Allah knows best.