ליה אדעת רבים to the priest based on the consent of the public, making it a type of vow that cannot be dissolved without their consent.
אמר אמימר הלכתא אפי' למ"ד נדר שהודר ברבים יש לו הפרה על דעת הרבים אין לו הפרה וה"מ לדבר הרשות אבל לדבר מצוה יש לו הפרה כי ההוא דמקרי דרדקי אדריה רב אחא דקא פשע בינוקי ואהדריה רבינא דלא אשכח דהוה דייק כוותיה: Ameimar says: The halakha is as follows: Even according to the one who says a vow that was taken in public has the option of nullification, if it was taken based on the consent of the public it has no option of nullification. The Gemara comments: And this matter applies only to when the nullification of a vow is in order to enable one to perform an optional matter, but to enable one to perform a matter of a mitzva it has the option of nullification. As in the incident involving a certain teacher of children,upon whom Rav Aḥa administered a vow to cease teaching based on the consent of the public, as he was negligent with regard to the children by hitting them too much; and Ravina had his vow nullified and reinstated him, as they did not find another teacher who was as meticulous as he was.
והמטמא למתים כו': מאי שנא הכא דסגי ליה בקבלה ומאי שנא התם דמדרינן ליה התם יצרו תקפו: § The mishna teaches that a priest who becomes impure through exposure to corpses is disqualified from performing the Temple service until he accepts upon himself that he will no longer become impure in that manner. The Gemara asks: What is different here that it is sufficient for him merely to accept upon himself a commitment in order to continue serving in the Temple, and what is different there, where the priest marries by a transgression, that we administer a vow to him? The Gemara responds: There his desire seizes him, and it is therefore necessary for the priest to take a vow.
הדרן עלך מומין אלו
מתני׳ יש בכור לנחלה ואינו בכור לכהן בכור לכהן ואינו בכור לנחלה בכור לנחלה ולכהן ויש שאינו בכור לא לנחלה ולא לכהן MISHNA: There is a son who is a firstborn with regard to inheritance but is not a firstborn with regard to the requirement of redemption from a priest. There is another who is a firstborn with regard to redemption from a priest but is not a firstborn with regard to inheritance. There is another who is a firstborn with regard to inheritance and with regard to redemption from a priest. And there is another who is not a firstborn at all, neither with regard to inheritance nor with regard to redemption from a priest.
איזהו בכור לנחלה ואינו בכור לכהן הבא אחר נפלים אע"פ שיצא את ראשו חי ובן תשעה שיצא ראשו מת המפלת כמין בהמה חיה ועוף דברי רבי מאיר Which is the son who is a firstborn with regard to inheritance but is not a firstborn with regard to redemption from a priest? It is a son who came after miscarriage of an underdeveloped fetus, even where the head of the underdeveloped fetus emerged alive; or after a fully developed nine-month-old fetus whose head emerged dead. The same applies to a son born to a woman who had previously miscarried a fetus that had the appearance of a type of domesticated animal, undomesticated animal, or bird, as that is considered the opening of the womb. This is the statement of Rabbi Meir.
וחכ"א עד שיהא בו מצורת אדם המפלת סנדל או שליא או שפיר מרוקם והיוצא מחותך הבא אחריהם בכור לנחלה ואין בכור לכהן And the Rabbis say: The son is not exempted from the requirement of redemption from a priest unless his birth follows the birth of an animal that takes the form of a person. In the case of a woman who miscarries a fetus in the form of a sandal fish or from whom an afterbirth or a gestational sac in which tissue developed emerged, or who delivered a fetus that emerged in pieces, the son who follows these is a firstborn with regard to inheritance but is not a firstborn with regard to redemption from a priest.
מי שלא היו לו בנים ונשא אשה שכבר ילדה עודה שפחה ונשתחררה עודה עובדת כוכבים ונתגיירה משבאת לישראל ילדה בכור לנחלה ואין בכור לכהן In the case of a son born to one who did not have sons and he married a woman who had already given birth; or if he married a woman who gave birth when she was still a Canaanite maidservant and she was then emancipated; or one who gave birth when she was still a gentile and she then converted, and when the maidservant or the gentile came to join the Jewish people she gave birth to a male, that son is a firstborn with regard to inheritance but is not a firstborn with regard to redemption from a priest.
רבי יוסי הגלילי אומר בכור לנחלה ולכהן שנאמר (שמות יג, יב) פטר רחם בישראל עד שיפטרו רחם מישראל Rabbi Yosei HaGelili says: That son is a firstborn with regard to inheritance and with regard to redemption from a priest, as it is stated: “Whatever opens the womb among the children of Israel” (Exodus 13:2). This indicates that the halakhic status of a child born to the mother is not that of one who opens the womb unless it opens the womb of a woman from the Jewish people.
מי שהיו לו בנים ונשא אשה שלא ילדה נתגיירה מעוברת נשתחררה מעוברת In the case of one who had sons and married a woman who had not given birth; or if he married a woman who converted while she was pregnant, or a Canaanite maidservant who was emancipated while she was pregnant and she gave birth to a son, he is a firstborn with regard to redemption from a priest, as he opened his mother’s womb, but he is not a firstborn with regard to inheritance, because he is not the firstborn of his father or because halakhically he has no father.
וילדה היא וכהנת היא ולויה היא ואשה שכבר ילדה וכן מי שלא שהתה לאחר בעלה שלשה חדשים ונשאת וילדה ואינו ידוע אם בן תשעה לראשון אם בן שבעה לאחרון בכור לכהן ואינו בכור לנחלה: And likewise, if an Israelite woman and the daughter or wife of a priest, neither of whom had given birth yet, or an Israelite woman and the daughter or wife of a Levite, or an Israelite woman and a woman who had already given birth, all women whose sons do not require redemption from the priest, gave birth in the same place and it is uncertain which son was born to which mother; and likewise a woman who did not wait three months after the death of her husband and she married and gave birth, and it is unknown whether the child was born after a pregnancy of nine months and is the son of the first husband, or whether he was born after a pregnancy of seven months and is the son of the latter husband, in all these cases the child is a firstborn with regard to redemption from a priest but is not a firstborn with regard to inheritance. Due to the uncertainty, he is unable to prove he is the firstborn of either father, and therefore he is not entitled to the double portion of the firstborn.