Gemara

(R. Yochanan): R. Shimon ben Gamliel and Chachamim argued about which is preferable, Sinai (one who knows all the Mishnayos and Beraisos), or one who uproots mountains (through his sharp power of reasoning).

Rav Yosef was Sinai, and Rabah uprooted mountains. They asked Chachamim of Eretz Yisrael, who answered 'Sinai comes first. All need the bearer of wheat.' (Mishnayos and Beraisos are the source of all Halachah.)

Eruvin 62b - Question (Abaye): The rule is, the Halachah always follows R. Eliezer ben Yakov. In this case, also Rav Yehudah taught that the Halachah follows him. May one give such a simple ruling in front of his Rebbi?

Answer (Rav Yosef): No. It is also simple that one may eat eggs with milk, yet Rav Chisda would not rule about this in the life of his Rebbi Rav Huna!

Rav Chisda ruled in Kafri while Rav Huna was alive.

R. Eliezer of Hegronya and Rav Acha bar Tachlifa visited Rav Acha brei d'Rav Ika. He wanted to slaughter for them. He showed the knife to them. Rav Acha said 'show it to Rav Acha bar Yakov. This is his region!' R. Eliezer said that Rava permits a Chacham to inspect his own knife! R. Eliezer inspected it.

R. Eliezer was punished. Even though Rava permits, since they wanted to bring it to the local Chacham, he should not have deterred them. Or, Rav Acha bar Yakov is different, for he was exceptionally aged and Chacham.

(Rava): One may stop someone from transgressing (even though it implicitly teaches a Halachah) in front of one's Rebbi.

Rav Ashi: "Ein Chachmah v'Ein Tevunah v'Ein Etzah l'Neged Hash-m" - we do not stop to honor the Rebbi when Hash-m's name is being desecrated.

Bava Metzia 33a (Beraisa - R. Meir): One's Rebbi is the one who taught him Chachmah, not the one who taught him written Torah or Mishnah;

R. Yehudah says, it is whom he learned most of his Chachmah from;

R. Yosi says, it is even if he explained only one Mishnah to him.

(R. Yochanan): The Halachah follows R. Yehudah.

(Rav Acha bar Rav Huna): The Halachah follows R. Yosi.

Bava Basra 145b (Beraisa): One who knows Agados is like one with much land and known possessions. A master of Pilpul (delving deeply into Torah) is like a moneychanger with coins and one who owns land that yields choice olives (he constantly profits). One who knows teachings is like one with wealth stored up. When the time comes, he profits. One who knows Gemara (to resolve difficulties of Mishnayos and Beraisos), he is like the one who sells wheat. Everyone needs him.

Yevamos 42b - Question (R. Nachum): If there is a Machlokes in the Mishnah and a Stam Beraisa, how do we rule?

Answer (R. Avahu): If Rebbi (who codified the Mishnah) did not decide the law, how could R. Chiya (who edited Beraisos) decide?!

Rishonim

Rambam (Hilchos Talmud Torah 5:3): If one was three Parsa'os from his Rebbi and someone asked a Halachah, he may answer. One may rule to separate from Isur even in front of his Rebbi. If one saw someone transgressing due to ignorance, or due to his evil, he may stop him and say 'this is forbidden', even in front of his Rebbi without Reshus (permission). We do not show honor to Rebbeyim if this will cause Chilul Hash-m. This refers to what happened by chance. One may not fix himself to rule for all who ask, even if he and his Rebbi are at opposite ends of the world, until his Rebbi dies or gives Reshus.

Rambam (9): ...This refers to Rabo Muvhak from whom he learned most of his Chachmah. If he did not learn most of his Chachmah from him, he is a Talmid Chaver. Even if he learned from him only one thing, be it big or small, he must stand for him and tear (his clothing when the Rebbi dies).

Hagahos Maimoniyos (40): Rashi explains that Rav Chisda ruled in Kafri while Rav Huna was in Pumbadisa.

Rosh (Bava Metzi'a 2:30): The Halachah follows R. Yehudah.

Rashi (5b DH k'Neged): Even though Machaneh Yisrael was three Parsa'os, everyone came to Moshe with questions.

Hagahos Maimoniyos (Hilchos Evel 9:1): Avi ha'Ezri says that we follow R. Yosi because Rav Sheshes, Rava and Ula hold like him, and they are Basra. Also, we generally rule like R. Yosi against R. Yehudah. Only R. Yochanan established the Mishnah in Bava Metzia like R. Yehudah.

Ritva (in Shitah Mekubetzes 33a): R. Yosi holds that even if he explained only one Mishnah to him, he is considered his Rebbi for everything.

Poskim

Shulchan Aruch (YD 242:4): If one was three Parsa'os from his Rebbi and someone happened to ask a Halachah, he may answer. He may not fix himself to rule for all who ask, even if he and his Rebbi are at opposite ends of the world, until his Rebbi dies or gives Reshus.

Rema: This refers to Rabo Muvhak. Even if he is not right in front of him, if they began with the honor of his Rebbi and said that they will ask him, or if the Rebbi is outstanding in Chachmah, one may not rule in his city.

Shulchan Aruch (ibid): Reshus from one Rebbi does not suffice. He must ask Reshus from all his Rabosav Muvhakim.

Source: Beis Yosef (DH u'Mashma, citing Maharik 169).

Maharik (169): Rabo Muvhak is the one who taught to him Halachos, even more than the one who taught to him Pilpul. In Yevamos, we say 'if Rebbi did not teach this, from where would R. Chiya (his Talmid) know it?!' This refers to deciding which Tana the Halachah follows. This depends on sharpness, not on Beki'us (breadth of knowledge). In Horayos, we say that Beki'us is more important than sharpness. This is not only to be the leader. The Gemara did not say that sharpness is better for deciding the Halachah! Also, there we discuss to be Rosh Yeshiva and to rule in practice.

Rema: Usually, Rabo Muvhak is the Rebbi who taught to him most of his Chachmah. Here, it cannot mean this, for if so, one cannot have more than one! Rather, here it means a proper Talmid, to exclude a Talmid Chaver, i.e. a Talmid who grew in Torah and became a colleague, i.e. almost as great a Chacham as his Rebbi.

Shach (12): The first opinion in the Rema holds that one is a Talmid even if he did not learn most of his Chachmah from him. I disagree. The Maharik says that Rav Achai Ga'on, Ri, Rambam, Rid, Rosh and Tur all rule like R. Yehudah, that one must honor only the Rebbi who taught to him most of his Chachmah. R. Chananel and Avi ha'Ezri rule like R. Yosi, but only for tearing and standing, but not for other laws of Rabo Muvhak. Maharik adds that R. Yochanan taught to Reish Lakish most of his Chachmah and gave Semichah to him. Even so, Reish Lakish argued with him and ruled in practice in front of him, for he became almost as great as him. The Rema's proof is invalid, for one can have many Rebbeyim Muvhakim, i.e. the ones who taught him most of his Mikra, Mishnah, Gemara, Agada, Kabalah, etc.

Rema: However, some distinguish and hold that if he received Reshus from his Rebbi, it helps to rule outside of three Parsa'os, but not within three Parsa'os. Some say that as long as he is not Rabo Muvhak, i.e. he did not learn most of his Chachmah from him, he is a Talmid Chaver.

Gra (14): Rashi holds that Reish Lakish learned primarily from R. Yochanan. He must say that Reshus helps (even in front of him).

Shulchan Aruch (11): One may separate from Isur even in front of his Rebbi. If one saw someone transgressing due to ignorance, or due to his evil, he may stop him and say 'this is forbidden.' We do not show honor to Rebbeyim if this will cause Chilul Hash-m

Chachmas Shlomo: Whether it is an Isur mid'Oraisa or mid'Rabanan, one may separate in front of his Rebbi. He need not do so.

Shulchan Aruch (30): Rabo Muvhak is the one from whom he learned most of his Chachmah, be it Mikra, Mishnah or Gemara.

Rema: Nowadays, the primary Rebbi does not depend on teaching Pilpul and the distinctions people make nowadays, rather, the one who taught to him Halachos and investigation and helped him attain what is Emes and straight.

Aruch ha'Shulchan (YD 242:19): This is like the Yerushalmi. R. Meir learned from R. Yishmael and R. Akiva. Rav Yehudah learned from Rav and Shmuel. Abaye and Rava learned from Rabah and Rav Yosef. In every case, both of these were considered Ravo Muvhak.