The twin prohibitions that biur chametz must address are already established by the Chazal themselves: the Mekhilta (Shemot 12:19) derives from two separate verses — "לֹא יִמָּצֵא" (Shemot 12:19) and "לֹא יֵרָאֶה" (Shemot 13:7) — that there are two distinct biblical prohibitions, bal yeira'eh and bal yimatzei, which together frame the entire obligation to eliminate chametz from one's possession.
The Gemara in Pesachim 21b adds that deriving benefit from chametz on Pesach is also scripturally prohibited, derived by Chizkiya from the passive formulation "לֹא יֵאָכֵל חָמֵץ" — meaning no permitted mode of use, including benefit, shall exist at all.
The foundational question of how bitul alone can discharge a Torah-level obligation is addressed by the Beit Yosef (Orach Chayim 431), who records that according to Rashi the source is the word "תַּשְׁבִּיתוּ" — since the Torah did not write "תְּבַעֵרוּ," removal through burning, it follows that hashbatah in the heart suffices; the Ran's view, also cited there, is that the verse "לֹא יֵרָאֶה לְךָ" teaches that chametz nullified in one's mind is no longer "yours," and that the two paths — bitul or physical search-and-destruction — are alternative means of fulfilling the same Torah command.
The Ramban (Shemot 12:19) grounds the entire structure of the prohibitions in the concept of reshut: the Torah forbids only chametz that remains in one's own domain, so chametz deposited with a non-Jew in the non-Jew's own home does not violate bal yeira'eh or bal yimatzei at all, and the Ramban notes that even rabbinic concern was not extended to it — a principle that undergirds the halachic logic of mechirat chametz.
The Divrei Chaim (Orach Chaim 1:8) presses this analysis further by arguing from the Rambam's own language that bitul is the Torah-level mechanism: the Rambam writes that the Torah's hashbatah is fulfilled when one nullifies chametz in his heart and considers it as dust, while the obligation to search and check is only rabbinic — and the Divrei Chaim reasons that this rabbinic requirement is itself contingent on the bitul having been performed, so that if one does not perform bitul, a Torah-level obligation to search would apply.
Regarding what the mechilah adds beyond bitul, the Shulchan Arukh (Orach Chayim 448:3) rules that chametz over which Pesach has passed is forbidden in benefit even if left accidentally or under duress — but that if it was given or sold to a non-Jew outside one's home before Pesach, even with the understanding that the non-Jew will guard it and return it afterward, it is permitted, provided the transfer was a complete and unconditional gift or a genuine sale, since a conditional transfer does not suffice. This establishes that a full legal transfer of ownership is indispensable: bitul alone cannot undo the chametz once Pesach has begun, whereas a valid mechilah removes it from one's reshut entirely.
The Tzemach Tzedek (Orach Chaim 47) reasons that since the non-Jew who has already entered into a sale arrangement has his own financial interest in remembering the chametz, there is no concern of forgetfulness invalidating the transaction — and he adds that in the current practice, where everyone sells through a single agent, the concern is even more remote, affirming the practical reliability of mechirat chametz as a complement to bitul.
Tzitz Eliezer (20:51) addresses, on our reading of the cited page, the custom of refraining from selling actual chametz — meaning high-value chametz she-be-ayin — even where a formal mechilah would technically suffice, reflecting a concern about the substantiveness of the sale in practice.
