• A היקש can be חוזר ומלמד with a kal vachomer; unresolved if it can be חוזר ומלמד with a בנין אב

Although the Gemara established that a היקש cannot be חוזר ומלמד with another היקש, nor with a gezeirah shavah, here it proves that it can be חוזר ומלמד with a kal vachomer, from a derashah of Rebbe Yishmael’s yeshivah (on Daf 41a), where the laws of "את בדם וטבילה" are derived from a hekesh with פר כהן משיח to the case of שעירי ע"ז, and then extended from שעירי ע"ז to the שעירי יום כפור with a kal vachomer. The Gemara asks if a hekesh can be חוזר ומלמד with a בנין אב. Rebbe Yirmiyah suggests a proof, arguing that the Torah did not need to teach the צפון requirement for אשם, and we could have derived it from חטאת with a בנין אב. However, this proof is rejected, because just as אשם could not be derived from עולה with a בנין אב, because of עולה’s unique stringency שכן כליל – that it is completely consumed on the מזבח, אשם can also not be derived from חטאת, since it is מכפר for a כרת violation (which אשם does not). The Gemara adds that none of these three could have been derived from the other two with a צד השוה.

• A גזירה שוה being חוזר ומלמד (machlokes if בתר למד or בתר מלמד if one is חולין)

The Gemara proceeds to discuss whether a גזירה שוה can be חוזר ומלמד with the four derashos. It first asks if it can be חוזר ומלמד with a hekesh, and Rav Pappa brings a proof that it can: a hekesh between שלמים and תודה teaches that a תודה can also be brought from maaser sheni money, like a shelamim, which shelamim itself had derived from a gezeirah shavah (שם שם) from maaser. However, Mar Zutra brei d’Rav Mari asked Ravina: מעשר דגן חולין בעלמא הוא – maaser of grain (this law’s original source) is mere chullin, and cannot prove the rules of derashos of קדשים!? Ravina replied: was the principle limiting two-step derashos of kodashim only said where למד קדש ומלמד קדש – both the derived law and the teaching case are kodashim?! It was not; rather, it applies wherever the למד – derived case is kodashim, so Rav Pappa’s proof, where the למד is שלמים, still stands. Mar Zutra, however, holds that if the מלמד – teaching case is חולין, the laws of two-step derashos do not apply.

• A kal vachomer being חוזר ומלמד (קו"ח בן קו"ח)

The Gemara proceeds to discuss whether a kal vachomer can be חוזר ומלמד with the four derashos. In its third question, it asks if it can be חוזר ומלמד with another kal vachomer. The Gemara seeks to resolve this question through its own kal vachomer: if a gezeirah shavah cannot learn from a היקש, yet it can teach through a kal vachomer (as the Gemara proved above), then a kal vachomer, which can learn from a היקש (as proven in the first point above), certainly can teach through another kal vachomer! וזהו ק"ו בן ק"ו – And this is a kal vachomer the “son” of a kal vachomer, i.e., this kal vachomer itself is based on another, since the a gezeirah shavah’s ability to teach though a kal vachomer was proven above through a kal vachomer. The Gemara objects: בן בנו של ק"ו הוא – It is actually a “grandson” of a kal vachomer, i.e., a triple kal vachomer (since the proof for one kal vachomer to teach with another is itself also a kal vachomer, and a derashah deriving a law with this method would be a kal vachomer). Perhaps a triple kal vachomer is not a valid derashah!? Therefore, the Gemara restructures its proof so that it is only a קו"ח בן קו"ח.

Siman – Nonstop Flight. The Kodshim crew on the nonstop flight going over the flight rules who weren’t sure if a rule learned from aהיקש can be חוזר ומלמדwith a בנין אב, and whether one goes בתר למד or בתר מלמד when one dish is חולין, were asked to quiet down so they don’t wake up the man wearing a kal v’chomer shirt and all the more so they don’t wake up his sleeping son wearing a קל וחומר shirt.