Gemara

(Beraisa): There are three partners in the formation of a child -- Hash-m, the father and the mother;

The father contributes white semen, which forms bones, sinews, (finger and toe) nails, the brain, and the white of the eyes;

The mother contributes red seed (blood), which forms skin, flesh, bones, hair, (blood) and the black of the eyes.

(Beraisa): At first, they used to say that the child is the opposite gender of the one who was Mazri'a (emitted his or her seed during relations) first. They did not explain the source;

R. Tzadok expounded "Eleh Bnei Leah Asher Yaldah l'Yakov... v'Es Dinah Vito." The verse attributes the sons to their mother, and the daughter to her father.

Chulin 70a - Question: Animals A and B had not yet given birth. Their wombs were stuck together. A male fetus left A's womb and entered B's. The child came out of B after the animals separated.

Clearly, the child is a Bechor, and the next child of A is not;

Does the child exempt also B's first child from Bechorah (since A's child was the first to leave B's womb)?

This question is not resolved.

Yevamos 97b (Beraisa): If two twins were conceived Lo bi'Kedushah (before the mother converted) and born bi'Kedushah, they do not do Chalitzah or Yibum, but they are liable for marrying each other's wife.

Rishonim

Rambam (Hilchos Isurei Bi'ah 14:14): If twin brothers were conceived Lo bi'Kedushah and born bi'Kedushah, they are liable for Eshes Ach.

Rambam (Hilchos Avadim 7:5): If one wrote to his pregnant Shifchah 'you are free and your fetus is a slave', his words are fulfilled. If he wrote "you are (still) a slave, and your fetus is free", his words have no effect, for this is like freeing half of her.

Rebuttal (Ra'avad): The Reisha contradicts the Seifa! The Seifa holds that Ubar Yerech Imo. If so, why are his words fulfilled in the Reisha?! Rather, in the Reisha his words are fulfilled because Ubar Lav Yerech Imo. In the Seifa his words have no effect because it is Lo Ba l'Olam, or because a slave cannot receive a Get Shichrur for another slave of his master.

Radvaz (1:188): The Rambam holds that Ubar Yerech Imo. The fetus is part of the mother, but the mother is not part of the fetus.

Rambam (Hilchos Bechoros 4:18): If one stuck two wombs together, and the fetus went from one to the other, it is a Safek whether the fetus exempts the latter animal from Bechorah, until its child is born.

Piskei Din (Yerushalayim Dinei Mamonos and Birurei Yuchsin 11 p.568): The Rambam says that it is not exempted from Bechorah until its child leaves its womb. I.e. the child inside is not considered its child, rather, the first animal's child. We can learn from here to a fetus transferred to another woman. However, we could say that the Gemara and Rambam discuss a large fetus uprooted from the womb and planted in another womb, but a tiny fertilized egg would not be attributed to the first mother. However, we do not distinguish. The Ri Korkus asked why the Rambam did not omitted the case of a Sheretz that swallowed a fetus and deposited it in another womb, and answered that such a case would never occur. It follows that the case of wombs stuck together could occur. This is unlike Tosfos (Kesuvos 4b DH Ad), who says that it would never occur. Nowadays we see that it is possible, but only with a small fetus. Therefore, the Rambam discussed a small fetus, and said that it is considered the child of the first mother.

Note: How he we learn from what doctors can do nowadays, to a fetus passing from one womb to the other (stuck to it) without outside help? I can imagine one womb forcibly pushing a sizable fetus into the other womb, but I cannot imagine how it could push a miniscule egg into another womb! If so, the Rambam discussed a larger fetus, but perhaps a fertilized egg is different!

Poskim and Acharonim

Shulchan Aruch (269:4): If twin brothers were conceived Lo bi'Kedushah (before conversion) and born bi'Kedushah, they are liable for Eshes Ach (if one marries the other's sister), but they do not do Chalitzah or Yibum.

Shach (6): The Beis Yosef and Bach say that this is even if they are not twins. This is difficult. The Mechaber connotes that it is only if they are twins, and so do the Levush, Rambam and Magid Mishneh (Hilchos Isurei Bi'ah 14:14).

Note: The Magid Mishneh merely says that the Rambam is from the Gemara. The Shach derives that he explains the Rambam simply, since he did not explain that the same applies even if they are not twins.

Dagul me'Revavah: Rashi (Yevamos 98a) holds that it is even if they are not twins. Also the Rambam does not require that they be twins. It suffices that they are from the same mother, even if the first was conceived Lo bi'Kedushah and born bi'Kedushah, and the latter was conceived and born bi'Kedushah. A proof is that the Rambam in Hilchos Melachim says that Tamar was Avshalom's maternal sister.

Piskei Din (ibid.): We learn from Yevamos that if a Yisrael's seed impregnated a Nochris, and then the fetus was put into a Yisraelis' womb, at the time of conception the seed was not attributed to the father. This is like a Yisrael who fathered a child from a Shifchah or Nochris. After the fetus was transferred, it is like a baby conceived Lo bi'Kedushah and born bi'Kedushah. The Shulchan Aruch rules that if twin brothers were conceived Lo bi'Kedushah and born bi'Kedushah, they are liable for Eshes Ach, but they do not do Chalitzah or Yibum. The Shach says that this is only if they are twins. He holds that birth itself does not make brotherhood. Therefore, they are considered brothers only if they were twins. However, Dagul me'Revavah and others consider them brothers even if they are not twins. He derives this from Rashi, and says that the Rambam agrees, if the latter was conceived bi'Kedushah. He holds that birth itself makes brotherhood. The Taz (267:30) holds that that there cannot be a slave inside a free woman. Therefore, he explains that the master frees the mother only after she gives birth. The Shach (267:75) and Radvaz hold that there cannot be a slave inside a free woman, so they explain the Rambam and Shulchan Aruch simply. The Shach holds that pregnancy and birth alone do not make Kurvah (kinship), even though he (and the Radvaz) hold that Ubar Yerech Imo. The Taz and other Poskim hold that Ubar Lav Yerech Imo. Even so, the Taz holds that birth determines Kurvah. If so, the latter woman is the mother. Surely we should not enter our heads into this argument between great Poskim, so we must forbid transferring a fetus or an unfertilized egg from one woman to another (for Chachamim required Havchanah, i.e. abstaining from relations when there could be uncertainty about the child's lineage).

Maharsha (31a DH v'Es): How can we expound from Dinah that when the father is Mazri'ah first, the child will be a female? We say that initially Dinah was a male, and through Leah's Tefilah, the fetus was switched to a male! I found written that Yosef was in initially in Leah's womb, and he was switched to Rachel's. I.e. she is attributed to Yakov for he was Mazri'a first (during relations with Rachel) in her conception.

Meshech Chachmah (Bereishis 46:22): Regarding Yakov's other wives, the Torah discusses the sons "Asher Yaldu l'Yakov." Regarding Rachel it says "Bnei Rachel Asher Yulad l'Yakov." Targum Yonason explains that Yosef was in initially in Leah's womb, and he was switched to Rachel's. The Gemara says that "Asher Yaldah" refers to conception; when she is Mazri'ah first, the child is a male. Since Rachel was not Mazri'ah to conceive Yosef, it does not say about her Asher Yaldu.

Meshech Chachmah (Bereishis 37:27): Yehudah said about Yosef "Acheinu Besareinu Hu." The Gemara says that the father's semen forms bones, sinews... and the mother's blood forms skin, flesh... Since Yosef was conceived in Leah's womb, he was "Acheinu" from the father, and "Besareinu" from the mother.

Mishneh Halachos (4:249): The Radvaz' son (in Teshuvos ha'Radvaz) learned from the Targum that if doctors would succeed to switch a fetus from one mother to another, the baby would be considered the child of the woman who gave birth to him, and not of the woman in whom conception occurred. One may not infer that it is permitted to do so.