שלום

א) במקרה שלנו הלימוד הוא אסמכתא בעלמא כפי שכתב רש"י בד"ה "זכר לדבר". ישנם כמה מקומות בש"ס עם לימוד בצורה דומה. למשל כתובות מ"ה: ובעירובין נ"א. באופן עקרוני בכל התורה כולה (חוץ מקדשים) למדים למד מן הלמד, כלומר שאפשר ללמוד בג"ש מראשון לשני, ומשני לשלישי וכן הלאה. דבר זה נתבאר בזבחים מ"ט:

בכבוד רב

יהודה לנדי

(a) Not only is each stage indispensable to the Limud, but it is also clear from the Gemara's question 'Mai ve'Omer?' that it also conforms with the rules of Hekesh and Gezeirah-Shavah. This is also borne out by Tosfos' question. And all this, despite the fact that, as the Gemara itself explains, this is not a real Limud, only an Asmachta, for the reason mentioned by Rashi ('Divrei Torah mi'Divrei Kabalah Lo Yalfinan'), in which case we may have thought that it doesn't really matter whether they follow the rules or not.

Incidentally, I thought at first, that in any event, it cannot be more than an Asmachta, since to begin with, Bedikas Chametz is only de'Rabbanan. However, since Rashi declines to say that, it appears he must hold that Bedikas Chametz is d'Oraysa (see P'nei Yehoshua and Tz'lach).

In view of the above, why should the multiple Limud not be acceptable? We often learn one independent Limud from another, outside the realm of Kodshim [see Temurah 21b, and the Sugya in Zevachim, cited in the Masores Hashas]. Similar teachings can be found in Eruvin 51a, Kesuvos 45b-46a. In Kesuvos it appears to be mid'Oraisa.

(b) As for your second question, the Ran seems to agree with Rashi, that the Tana'im are at one with R. Chisda, since in his comment on the Rif who only cites the second Beraisa, he refers to R. Chisda's explanation. Indeed, it seems clear that this is so from the fact that both Beraisos cite the same four Pesukim as he doe sand even in the same order.

The reason the Gemara finds it necessary to cite all three sources is because each one teaches us something that we would not have known from the one before it. R. Chisda teaches us the source for Bedikas Chametz be'Ner from the Pesukim concerned, Tana de'Bei R. Yishmael shows that the source is from the Tana'im, and that it is only a Zecher, not a proof. The second Beraisa clarifies precisely what 'Ner' comes to preclude ('Or ha'Chamah, Or ha'Levanah and Or ha'Avukah').

(Presumably Rav Chisda was not aware of the Beraisos, since he would have quoted them had he known of them.)

Eliezer Chrysler.