This is a well proven true narration from Sunni sources, Imam Tirmidhi also wrote this. It clearly says that "Ali is from me and I am from Ali, No one can convey on my behalf except myself or Ali".
We can also note that Allah(swt) and his Prophet (pbuh) did not give Abu Bakar the responsibility to recite some verses of Surah Taubah, how can it be possible for Abu Bakar to have authority to interpret the whole Quran?
English Narration
Habshi ibn Janadah Salooli narrated:
The Holy Prophet said:
“Ali is from me and I am from Ali and no one can discharge my duty except myself or Ali.”
Another narrator narrates
English Translation
Saad Narrated:
The Holy Prophet send Abu Bakr with Surah Taubah he went little far away then the Prophet send Ali. Ali get back Surah Taubah from Abu Bakar and proceeded to Mecca. Abu Bakar did not like this, the narrator said Holy Prophet said to him: “This work nobody will carry except myself or a person from my Ahlulbayt”.
Another Narration by Ali ibn e Abu Talib(as)
(Imam) Ali Ibn e Abu Talib narrated:
The Holy Prophet sent Abu Bakar towards Mecca with Surah Taubah. Then the Prophet sent me (Ali) after him to get Surah Taubah from him, when Abu Bakar returned back , he was not happy, he (Abu Bakar) asked the Holy Prophet, “is there any revelation about me (that Surah Taubah was given to Ali instead of me)?” The Holy Prophet said “I am given orders that I should take that Surah myself or the person (Ali) from my Ahlalbait.”