Challah is the small amount of dough that must be separated and given to a kohen. It is much like terumah and only a kohen can consume the challah provided it remains tahor. The Mishnah (4:2) teaches that a woman that is a tevul yom can prepare dough and even separate some of the dough for challah. She can then place that small amount in a separate kli and bring it close to the dough and designate it challah. From that point onward, since it is now defined as challah, she cannot handle it otherwise she would make the challah pasul. That is because a tevul yom is considered a sheni le'tumah. While the tevul yom cannot affect regular food, it does however invalidate terumah and challah. The novelty of the Mishnnah is that prior to separating challah, even though challah will need to be removed from the dough, the entire mass has the status of regular food and cannot be affect by a sheni le'tumah.
The next Mishnah (4:3) appears to teach the same law. The case and ruling are affectively the same. The only difference is that instead of the woman being a tevul yom, it is the kneading bowl. The Mishnah similarly teaches that the bowl can be used to knead the dough provided that when the challah is designated, it is no longer in contact with the kneading bowl. Why is this second Mishnah needed?
The Bartenura explains that this Mishnah is needed since one might have thought one needs to be stricter with a tevul yom vessel out of concern that, if permitted, one might then use a tameh vessel.
The Tosfot Yom Tov cites the Maharam who also explains that one might have thought to treat the case of a tevul yom bowl differently. He however explains that it is out of the concern that one might forget and designate the challah while it is still in the bowl. It is natural for one to be more careful if they are tameh as compared to other objects. The Mishnah therefore teaches that the tevul yom kneading bowel can still be used.
The Mishnah Achrona understands that the novelty of this Mishnah is that the solution is different. In the previous Mishnah the separated dough had to first be placed in the chafisha -- a vessel that is not susceptible to tumah (Barternua). In our Mishnah however there is no mention of the chafisha. The Mishnah Achrona explains that when the person is tameh the dough needs to be placed in the separate kli to prevent the challah from becoming tameh. In this case however, the person is tahor. Consequently, that dough can be placed nearby or even on top of the remaining dough when declaring it as being challah. That is because the remaining dough will be tahor even though it is contact with the bowl.1
We see both these alternative solutions in two other mefarshim. The Rashash understands that this case is different from the previous Mishnah in that she can hold the portion of dough when declaring it challah. The Chiddushei Mahariach however understands that the novelty of the Mishnah is that one can separate a portion and then let it rest on the remainder when declaring it Challah. The novelty is not just that is a possible solution, since, as explained above, the challah will not become tameh in the process. It is rather that one might think that one should nonetheless avoid separating challah this way out of concern that the challah might then touch the bowl. The Mishnah therefore teaches, that separating in this manner is sufficient and one need not take extra precautions.
1 The Mishnah Achrona however explains that the dough must still be separated. If however a portion was designated as challah without first separating it, the challah and the remaining dough would be considered connected (chibur) such that the challah would be considered touching the bowl directly.
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