Gemara

(Rava): If a pregnant cow damaged, one may collect from the calf.

This is because a fetus is (part of) the same body as its mother.

Temurah 25a (R. Yochanan): If one was Makdish a pregnant animal for a Chatas and it gave birth, he may bring either for his Chatas. (He holds that Ubar Lav Yerech Imo, i.e. a fetus is not like a limb of its mother. It is like another animal.)

Question (Beraisa): If one gave a document of freedom to his pregnant Shifchah and said "you are (still) a slave, and your fetus is free", she acquires for her child.

According to the opinion that Ubar Lav Yerech Imo, she cannot acquire for her child (another slave), for the Get never left her master's Reshus!

R. Yochanan is refuted.

Yevamos 78a (Rava): If a pregnant woman converted, her child need not immerse. (Her Tevilah counts for him.)

Sanhedrin 80a (Beraisa): If a cow gored and gave birth before it was sentenced, the calf is permitted (even if it was pregnant at the time of the goring)!

Question: Rava disqualifies (for Korbanos) the calf of a cow that gored or had Bi'ah with a man. The calf was also party to the goring or bestiality!

Answer #1 - Correction (Beraisa): Rather, if a cow gored, became pregnant and gave birth before it was sentenced, the calf is permitted. If it became pregnant and gave birth after it was sentenced, the calf is forbidden.

Question: This is according to the opinion that forbids Zeh v'Zeh Gorem (something that results from two or more causes, one of which is forbidden). How can we answer for the opinion that permits Zeh v'Zeh Gorem?

Answer #2 (Ravina): Rather, the Beraisa permits if it gave birth before it was sentenced. If it was sentenced and then gave birth, the calf is forbidden, for Ubar Yerech Imo (it was sentenced with its mother).

Chulin 58a (Mishnah - R. Eliezer): The child of a Tereifah is Pasul (for a Korban);

R. Yehoshua says, it is Kosher.

(Rav Acha): They argue about when the mother became Tereifah, and then it became pregnant. R. Eliezer forbids Zeh v'Zeh Gorem, and R. Yehoshua permits.

(Ravina): They argue about a pregnant animal that became Tereifah. R. Eliezer holds that Ubar Yerech Imo; R. Yehoshua holds that Ubar Lav Yerech Imo.

The argument applies even to Chulin. They argue about Korbanos to show the extremity of R. Yehoshua's opinion. He permits the child even for a Korban.

74a (Mishnah - Chachamim) If one slaughters an animal and finds inside a live Ben Tesha (a 9-month viable fetus), Shechitah of the mother permitted the child;

R. Shimon Shezuri says, even if the child is five years old and plowing in the field, Shechitah of the mother permitted it.

75b - Question: R. Shimon Shezuri holds like Chachamim!

Answer (Rav Kahana): They argue if the child stepped on the floor. (Chachamim require Shechitah, lest people come to eat other animals without Shechitah.)

75a (R. Ami): If one is Shochet a Treifah and finds a full term fetus inside, R. Meir permits the fetus through Shechitah. Chachamim forbid, even if it is slaughtered.

(Rava): All permit this fetus through its own Shechitah. The Torah permits a fetus through Shechitah of its mother (if she is Kosher) or through its own Shechitah.

(Rav Chisda): The child must be slaughtered.

Beitzah 6a (Rav): If a chick hatched on Yom Tov, it is forbidden.

Question (Rav Kahana and Rav Asi): Why is this different than a calf born on Yom Tov to a Tereifah cow? (The mother is Muktzah, but the child is permitted!)

Rishonim

Rambam (Hilchos Shechitah 12:10): One may slaughter a pregnant animal. Ubar Yerech Imo. If the fetus came out alive after slaughter and stepped on the ground, we do not slaughter it the same day. If one did, he is not lashed.

Source (Kesef Mishneh): This is from the Tosefta (Chulin 4:2).

Radvaz: In many places we say Ubar Yerech Imo. If it stepped on the ground, we require Shechitah mid'Rabanan.

Question (Tosfos Sanhedrin 80b DH Ubar): Our Gemara explained according to the opinion that Ubar Yerech Imo, and Zeh v'Zeh Gorem Asur. It asked how the opinion that permits Zeh v'Zeh Gorem can answer, but not for the opinion that Ubar Lav Yerech Imo. This shows that the Halachah is Ubar Yerech Imo. Perhaps our Gemara merely sought to answer for Rava. However, we can learn from him! In Chulin, Tana'im argue about Ubar Yerech Imo according to the opinion than a Tereifah cannot give birth, and this is the Halachah. The Halachah follows R. Yehoshua (who says that Ubar Lav Yerech Imo), for R. Eliezer is from Beis Shamai. Beitzah 6a permits the child of a Tereifah!

Answer #1 (Tosfos and Rosh Bava Kama 5:2): Ubar Yerech Imo in every respect, except for Tereifah. Since it has its own life, we do not say that it became Tereifah along with its mother.

Tosfos (ibid.): In Temurah, the Gemara refuted R. Yochanan, then suggested that Tana'im argue about this. The Gemara often refutes an opinion that is not primary, even though it could say that some Tana'im hold like it. We challenged Rava from R. Yochanan (Yevamos 78a) to show that all agree with Rava's law.

Answer #2 (Tosfos, citing Rav R. Tam): Everywhere, the Halachah is Ubar Lav Yerech Imo. The text in Temurah should not say that R. Yochanan was refuted. Rava says that a sentence passed on a pregnant animal applies also to the fetus, because until now its life depended on its mother, not because Ubar Yerech Imo. Likewise, we do not wait for a woman to give birth before executing her. However, it is difficult to explain why Rava says that one can collect from the calf to pay for the mother's share of the damage, unless we say Ubar Yerech Imo.

Poskim

Shulchan Aruch (YD 16:10): One may slaughter a pregnant animal. Ubar Yerech Imo. If the fetus came out alive after slaughter and stepped on the ground, we do not slaughter it the same day. If one did, he is not lashed.

Shulchan Aruch (79:3): An animal born to a Tereifah animal is Kosher.

Beis Yosef (DH veha'Nolad): We learn from Beitzah 6a and Chulin 75a. The Rashba says that even though Ubar Yerech Imo, we learned that a Tereifah cannot live. Therefore, the Isur of the mother is only because she cannot live. The fetus can live. Its life does not depend on its mother, so it is permitted.

Shach (8): We discuss when the mother became pregnant before it became Tereifah. If not, a Tereifah animal cannot give birth (if the mother was Safek Tereifah, the birth would prove that it is Kosher). Alternatively, we discuss a Vadai Tereifah mother. Even though an egg of a Tereifah bird is forbidden, that is because it grew b'Isur.

Taz (5): R. Feivish was unsure about the fetus if the mother was scratched by a animal with venom in its claws.

Gra (5): This is like R. Yehoshua (Chulin 58a), for the Gemara in Beitzah and Chulin 75a are like him.