Were there non-Imam Prophets as per Shia Islam?

Question

From Al Islam, we have:

The Shi’a believe that the rank of Imamat (the position of a divinely-appointed leader) is higher than that of prophethood and messengership.

... Prophet Muhammad (S) is has the highest rank among mankind, and is the most virtuous creature of God and the most honored before Allah. The above belief does not undermine his position since Prophet Muhammad was an Imam during his time as well!

With the possibility of being both Prophet and Imam, it seems that in order to meaningfully consider the rank of Imam higher than the rank of Prophet, there would need to be non-Imam Prophets (otherwise all Prophets would be Imams anyway). Hence my question:

Question: Were there non-Imam Prophets as per Shia Islam?

I'm not sure if this is possible, because the Qur'an (2:136 and 3:84) instructs us to not distinguish between Prophets. If there were non-Imam Prophets, then Prophet Muhammad would have higher rank, thereby distinguishing between Prophets.

Answer 1

Let me dissect your questions:

Were there non-Imam Prophets as per Shia Islam? AFAIK only Prophet Ibrahim and Prophet Muhammad were also Imams.

Does the Qur'an instruct us to not distinguish between Prophets? In terms of their message their is nothing different about them. But are some prophets higher in rank?

There are verses in Quran speaking nobly of the house of a prophet e.g. ال ابراهیم or ال عمران .

Nisa' 54:

أَمْ يَحْسُدُونَ النَّاسَ عَلَىٰ مَا آتَاهُمُ اللَّهُ مِن فَضْلِهِ ۖ فَقَدْ آتَيْنَا آلَ إِبْرَاهِيمَ الْكِتَابَ وَالْحِكْمَةَ وَآتَيْنَاهُم مُّلْكًا عَظِيمًا

Or do they envy people for what Allah has given them of His bounty? But we had already given the family of Abraham the Scripture and wisdom and conferred upon them a great kingdom.

Al-Imran 33

إِنَّ اللَّهَ اصْطَفَىٰ آدَمَ وَنُوحًا وَآلَ إِبْرَاهِيمَ وَآلَ عِمْرَانَ عَلَى الْعَالَمِينَ

Indeed, Allah chose Adam and Noah and the family of Abraham and the family of 'Imran over the worlds

_{the Family of Abraham and Imran have been better families than Adam and Noah}

Baqara 253:

تِلْكَ الرُّسُلُ فَضَّلْنَا بَعْضَهُمْ عَلَىٰ بَعْضٍ ۘ

Those messengers - some of them We caused to exceed others...