All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
As for the expression 'bi-ithnillah', if it is intended to mean the universal predestinate permission, then it is synonymous with the expression 'if Allah wills'.
Ibn ‘Uthaymeen was asked in the Open Door Meeting, “Is it permissible to make Istithnaa' when making an oath with other than the expression 'if Allah wills' by, for example, saying bi-ithnillah?"
He replied, “Yes, it is permissible; "bi-ithnillah" is like "Allah willing.""
As regards the ruling of making an exception with Allah willing or bi-ithnillah or the like in supplication, then Abu Hurayrah narrated that the Prophet said, “When one of you makes a supplication (to his Lord) he should not say 'O Allah, grant me pardon, if You so will'; rather, he should ask (his Lord) with determination and ask for great things, for there is nothing too great for Allah to give.” [Al-Bukhaari and Muslim]
This prohibition – which is a matter of difference of opinion as to whether it denotes dislike or forbiddance – is when the supplication is in the imperative form (i.e. a command), such as saying 'O Allah, do this for me, if You so will' or 'if You give permission'.
However, if the supplication is in the form of a statement, such as by saying 'so-and-so is victorious, Allah willing' or 'so-and-so is forgiven, bi-ithnillah', then this is not included in the prohibition. Ibn ‘Abbaas narrated that whenever the Prophet visited an ailing person, he would say, "La ba's, tahoorun in shaa' Allah (meaning: No harm, you will be purified, if Allah wills)." [Al-Bukhaari]
Ibn ‘Umar narrated that when the Prophet broke the fast, he would say, “Thirst has gone, the veins have become moistened, and the reward is confirmed in shaa’ Allah (Allah willing).” [Abu Daawood - Al-Haakim classified it as Saheeh (sound)]
The saying of the Prophet “tahoorun in shaa' Allah” and “the reward is confirmed in shaa’ Allah” are supplications, yet the Prophet made them conditional upon the Will of Allah because they are in the form of a statement (not in the imperative form).
Dr. Abdul-Kareem Al-Khudhayr said:
“If the supplication is in the form of a command, like 'O Allah, forgive me; grant me mercy; O Allah, forgive so-and-so', then it is not permissible to make it conditional upon the Will of Allah, and there is a prohibition on this. But if it is in the form of a statement, even if it is a supplication, then there is no harm to say, 'May Allah forgive you, in shaa’ Allah', or 'Jazaaka Allahu Khayran (may Allah reward you), in shaa’ Allah', because the Prophet said, 'Tahoorun in shaa' Allah,' and he said, 'The reward is confirmed, in shaa’ Allah,' when he broke the fast. There is nothing wrong in this if it is a statement and not in imperative form.” [Excerpted from Sharh Sunan At-Tirmithi]
For more benefit, please refer to fatwa 341359.
As regards a person who does not know the ruling of making Istithnaa' with bi-ithnillah, then ignorance of the Islamic ruling prevents holding someone accountable for his deeds, in general, and a person is acquitted.
Ibn ‘Uthaymeen said:
“There are impediments for rendering someone accountable, among which are ignorance, forgetfulness, and coercion, as the Prophet said, 'Allah has forgiven my nation for things they forget, or do by mistake, or which they are forced to do.' [Ibn Maajah and Al-Bayhaqi] Evidence from the Quran and the Sunnah prove the authenticity of this hadeeth. Ignorance means not knowing, so if an adult person does something forbidden because he does not know about its prohibition, then he is not held accountable for it.” [Excerpted from Majmoo’ Al-Fataawa]
Allah knows best.