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These are two concepts in Judaism:

עשה דוחה לא תעשה - that if there is a Mitzvah Asei in the Torah, and a Mitzvas Lo Sasei, we say that the Asei overrides the Lo Sasei, and one should do it.

For example: Attaching woolen Tzitzis to a Linen garment.

מצווה הבאה בעבירה - that if there's a Mitzvah and one does it "through" an Averiah, we say that it's not a Mitzvah, and one should not do it.

For example: Eating stolen Matzah on Pesach.

These two ideas seem similar. What makes us decide where to apply the 1st idea vs. the 2nd? What's the underlying difference?

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The principle of עשה דוחה לא תעשה tells one how to act when there two conflicting imperatives or obligations from the Torah. For example on one hand the Torah is commanding you to attach wool tzitzit to the four cornered garment. On the other hand the Torah is commanding you not to mix wool and linen. These are two conflicting imperatives that you must reconcile. It is the obligation itself that places you in the situation of conflict. Halacha provides a method of reconciliation. Perhaps it is really reconciling an internal conflict in the system itself.

In the situation of a מצוה הבאה בעבירה it is not two conflicting imperatives that place you in a situation of conflict. The Mitzvah can be done in different manners. You choose to fulfill the mitzvah in a manner that violates one of the prohibitions in the Torah. Parenthetically, even if you feel the violation is the only way to complete the performance, such as a situation where you have no Matzah, Halacha provides a category of Ones Rachmonei Patrei (if it is beyond your control you would be exempt from the performance).

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Tzitzis is also something that can be done without sha'atnez, but is allowed to be done anyhow. I believe a major component of עשה דוחה לא תעשה is that it is something in which both the aseh and lo ta'aseh are happening simultaneously, which is not so for מצוה הבאה בעבירה. –  jake Jun 24 '11 at 7:16
    
Jake is correct; aseh docheh is only when they occur simultaneously. –  Shalom Jun 24 '11 at 14:03
    
Couldn't the Aveirah and the Mitzvah occur simultaneously as well? –  RCW Jun 24 '11 at 16:08
    
For example if one picked up someones Lulav to fulfill the mitzvah. So he stole at the same time as fulfilling the Mitzvah of Lulav. Wouldn't that be simultaneous? Or if one ate Matzah that was Tevel or any other prohibited eating. –  RCW Jun 24 '11 at 16:15
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@jake, the mitzva couldn't be performed with this garment without trangressing; therefore, it is unavoidable. Tosfos Kesubos ch. 3. –  Barry Jun 24 '11 at 17:11
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If all conditions are right for the Asei to override the Lo Sasei, then it is permitted. Otherwise, it is a Mitzva haBa bAveira.

Among the conditions that must be met:

  • Fulfillment of the mitzva must occur at the same time (or with the same action) as the Aveira
  • It isn't possible to avoid the conflict by performing the mitzva some other way
  • The Aveira mustn't be accompanied by another Mitzva
  • The Aveira cannot be a serious one; e.g. punished by Kareis
  • The situation doesn't involve the Temple
  • The Asei isn't evadable via another person's forgiveness
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What about being makriv a korban with a mum? I thought that in the Bar Kamtza story they should have brought it anyway. –  Adam Mosheh Jun 11 '12 at 18:07
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R' Shimon Shkop (Nedarim, siman 1), if I remember correctly, answers that we use aseh docheh lo saaseh only when it doesn't negatively impact someone else (e.g. stealing).

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