Which Mitzvot take precedence?
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There are a number of principles that determine which Mitzvah takes precedence over another Mitzvah whenever two Mitzvot come into conflict.
The more common Mitzvah
- One of the major principles is Tadir VeSheino Tadir, Tadir Kodem; the more frequent of the two mitzvot takes precedence. 
- Examples of this principle can be found in Kitzur Shulchan Aruch 10:2 regarding putting on Tallit before Tefillin, Kitzur Shulchan Aruch 76:12 regarding saying Mincha before Mussaf, and Kitzur Shulchan Aruch 139:14 regarding the Torah reading for Chanukah and Rosh Chodesh.
- A mitzvah that applies directly to oneself. For example, a father and his son have yet to be redeemed from a cohen(pidyon haben); in the situation where the father only has enough to redeem himself or his son, he is to redeem himself and then his son. 
- A passing mitzvah. For example, when faced with the choice of performing a mitzvah that may be done only within the next five minutes versus one that may be done over the next hour, we are to perform the mitzvah that applies immediately and move on the next.
Learning Torah and Other Mitzvot
- While learning Torah is one of the most important mitzvot if it comes into conflict with a mitzvah that only you can do, the mitzvah takes precedence.
- One explanation for this halacha is that a person's learning should lead up to fulfilling the mitzvot and if a person abandons fulfilling mitzvot in order to learn Torah, it indicates that one's learning isn't done in order to fulfill the mitzvot.
- Another explanation for this is that learning Torah is all encompassing and really one should learn all of the time. If one wouldn't interrupt learning in order to do other mitzvot, one would never get a chance to do any mitzvot which Hashem commanded.
- In a general sense, learning Torah is more important than Kibbud Av VeEm.
Middot versus Mitzvot
- It is better to give up a mitzvah so others can do it rather than fight over it.
- It is worth fighting to be able to do a mitzvah if otherwise no one will do the mitzvah.
- For the source of this halacha, see Gemara Brachot 27a, 51b, Pesachim 114a, Megillah 29b, Sukkah 54b, and Zevachim (Mishna 10:1).
- Kiddushin 29a
- Kiddushin 29b, Berachot 27a
- Mishna Peah 1:1
- Gemara Moed Katan 9b, Rambam (Talmud Torah 3:4)
- Rabbenu Yonah (Brachot 9b at the top)
- Chida in Chaim Shaal 1:71:2, Gra in Sadeh Eliyahu Peah 1:1
- Gemara Megillah 16b, Rambam (Mamrim 6:13)
- Yoma 39a, Binyamin Zeev 163, Darkei Moshe 53:11, Magen Avraham 53:26, Mishna Brurah 53:66. Machasit Hashekel 53:26 based on Rashi and Tosfot Yeshanim learns that only if one stands to lose out on a partial mitzvah one shouldn’t get involved with something that could lead to a fight. But if he is going to lose out on a real mitzvah he shouldn’t get involved if it is definitely going to cause a fight. The Mishna Brurah understands the gemara to mean even if you’re going to lose a real mitzvah it is worth it not to fight.
- Mishna Brurah 53:66