Dedicate Daf 30a to:

שלישי הבא מחמת שני דשני גופיה אסור בחולין אינו דין שעושה רביעי בקודש
All the more so should a
Sheini caused Shelishi
be able to
make sanctified food into a Reviyi.

The Gemara is concluding the proof started on page . The Gemara there is figuring out if Rabbi Yossi follows Abba Shaul's opinion that a Tevul Yom can make food impure through a proxy or if he follows the opinion of the Sages that a Tevul Yom can only make food impure directly.
The Gemara says that if Rabbi Yossi follows Abba Shaul, he would derive the law that sanctified food can become a Reviyi from the law of the Tevul Yom and not from the .
Rather, Rabbi Yossi must follow the opinion of the Sages. If so, the Gemara concludes .
The Gemara asks:
וכי תימא משום דאיכא למיפרך מה לטבול יום שכן אב הטומאה
And if one will respond that a Tevul Yom
come from an Av HaTumah,

And is stronger than a regular Sheini, so perhaps only a Tevul Yom can make a Reviyi, but not another Sheini.
The Gemara answers:
הא אייתינה ממחוסר כיפורים ולא פרכיה
Rabbi Yossi used
a Mechusar Kippurim
to prove that sanctified foods can become a Reviyi, and wasn't concerned that a Mechusar Kippurim may have originally been an Av HaTumah.

Rabbi Yossi can answer that a Tevul Yom
a regular Sheini, as the original (Av HaTumah) impurity was removed when he went to the Mikvah, and the " Tevul Yom " impurity is a new, temporary, Sheini impurity.
אמר רב אסי אמר רב
Rav Assi said in the name of Rav,

ואמרי לה אמר רבה בן איסי אמר רב
And some say that Rabbah ben Assi said in the name of Rav:

ר"מ ור' יוסי ור' יהושע ור' אלעזר ור' אליעזר כולהו סבירא להו דאין שני עושה שלישי בחולין
Rabbi Meir, Rabbi Yossi, Rabbi Yehoshua, Rabbi Elazar, and Rabbi Eliezer all held that a Sheini cannot make non-sanctified
food into a

The Gemara will now prove it:
The source that
Rabbi Meir
holds that a Sheini cannot make non-sanctified food a Shelishi

דתנן כל הטעון ביאת מים מדברי סופרים מטמא את הקודש ופוסל את התרומה ומותר בחולין ובמעשר דברי ר"מ
Rabbi Meir says
(in a Mishna)
that one who is Rabbinically
impure (and makes food impure on a Sheini level)
makes sanctified food impure
enough that this food can make
food impure (to a
, makes Terumah impure
(to a
, and may eat non-sanctified food and Maaser

וחכמים אוסרין במעשר
The Sages, however, forbids
one Rabbinically impure
to close
But, as Rashi points out, he may touch
eat close
But even they agree that Rabbi Meir that one Rabbinically impure does not make non-sanctified foods into a Shelishi.

ר' יוסי הא דאמרן
The source that
Rabbi Yossi
holds that a Sheini cannot make non-sanctified food a Shelishi

דאם איתיה לייתיה לרביעי בתרומה וחמישי בקודש
Because if
a Sheini could make non-sanctified food a Shelishi
, Terumah would be able to become a Reviyi and sanctified food would be able to become a Chamishi.

The proved that Terumah is always able to go one step lower than non-sanctified foods, and sanctified foods one step lower than
As the Gemara says that Rabbi Yossi limits Terumah to a Shelishi and sanctified products to a Reviyi, he must hold that regular food cannot get lower than a Sheini.
רבי יהושע
The source that Rabbi Yehoshua
holds that a Sheini cannot make non-sanctified food
a Shelishi:

The Mishna says

ר"א אומר האוכל אוכל ראשון ראשון
Rabbi Elazar said that one who eats a Rishon becomes a Rishon,

שני שני
One who eats a
a Sheini,

שלישי שלישי
One who eats a
a Shelishi.

These impurities are Rabbinic enactments, as they were concerned that one will drink Terumah wine while eating impure foods, rendering the wine impure in his mouth.
רבי יהושע אומר האוכל אוכל ראשון ואוכל שני שני שלישי שני בקודש ואין שני בתרומה
Rabbi Yehoshua said that whether one eats a Rishon, a Sheini, or a Shelishi, he becomes a Sheini
sanctified foods, but does not
a Sheini for Terumah,

בחולין שנעשו על טהרת תרומה
This Shelishi must be
unsanctified food which the owner treats as

על טהרת התרומה אין על טהרת הקודש לא
One eating a Shelishi becomes a Sheini only when the food is meant to be treated as
Terumah, not as
regular, unsanctified food, as regular food does not become a Shelishi.

A Kohen may decide to eat all of his food in the type of purity required for Terumah, but not for sanctified food, and the decision to treat regular food as sanctified is null.
Therefore, a Kohen who regards his regular food as Terumah may sometimes end up eating Shelishi food. However, regular food which he treats as sanctified food (which is, halachically, regular non-sanctified food) does not make him a Sheini, as ...
אלמא קסבר אין שני עושה שלישי בחולין
food cannot become a Shelishi.

The source that Rabbi Elazar
holds that a Sheini cannot make non-sanctified food
a Shelishi:

As a Braisa

ר' אלעזר אומר שלשתן שוין
Rabbi Elazar said
that the following
impurities have
the same

הראשון שבקודש ושבחולין ושבתרומה מטמא שנים ופוסל אחד בקודש
A Rishon of sanctified
, unsanctified
, and Terumah can make sanctified
to three levels

מטמא אחד ופוסל אחד בתרומה
A Rishon of sanctified food, unsanctified food, and Terumah
can make Terumah impure
to two levels

ופוסל אחד בחולין
A Rishon of sanctified food, unsanctified food, and Terumah
can make
unsanctified food
but the impurity cannot spread

So regular food can only become a Sheini, and not a Shelishi.
ר' אליעזר
The source that Rabbi Eliezer
holds that a Sheini cannot make non-sanctified food
a Shelishi:

A Mishna

ר"א אומר חלה ניטלת מן הטהורה על הטמאה
Rabbi Elazar said that Challah can be taken from
dough to free ritually

The issue is that Challah (as well as all gifts separated for the Kohen) must be taken from a batch (so one is not allowed to separate Challah from one dough to free another dough). Normally, one would bring the two batches together (uniting them), and separate one portion of Challah from both batches simultaneously. The issue is that Challah should be separated in purity, so that a Kohen may eat of it, and the moment the two batches (one of which is impure) touch, they become impure. So the Gemara will explain how to fulfill both requirements (that the Challah should be taken from a batch and should be pure):

שתי עיסות אחת טהורה ואחת טמאה
there are two batches of dough, one is pure and one is not,

נוטל כדי חלה מעיסה שלא הורמה חלתה
the amount of dough required for
Challah from
the pure batch
(which did not have its Challah taken yet),

ונותן פחות מכביצה באמצע
And puts less than an egg size in the middle
(touching both batches)

And since food less than the size of an egg does not spread impurity, that bit will not spread impurity to the first (pure) batch.
כדי ליטול מן המוקף
So that one takes
from a batch.

The procedure is to have the three batches of dough touching each other in a daisy-chain :
1. The impure dough,
2. A piece of dough smaller than the size of an egg (which becomes impure, but doesn't spread impurity),
3. The pure dough (which stays pure).
The Challah, which is then taken from this third (pure) dough, is pure and may be given to a Kohen.

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