Bechorot 46 - Firstborn For Inheritance But Not for Kohen
Some blemishes apply only to animals but do not prevent a Kohen from serving in the Temple: if a calf mother was sacrificed that day, one cannot bring her offspring, but a Kohen and his son can serve on the same day; an animal that is terefah (sick and therefore not kosher), one delivered by Caesarean section, and one with whom bestiality was committed - all cannot be brought as sacrifices, but these situation do not disqualify a Kohen. A Kohen married to a woman prohibited to him cannot serve until he vows to divorce, and likewise one who visits cemeteries.
A child can be a firstborn for inheritance but not a firstborn for a Kohen - that is, he does not need to be redeemed by giving five silver coins to a Kohen. How so? If a woman was pregnant with twins, and the head of one emerged first, but then retracted and he died, and the second child was born healthy. The firstborn for a Kohen is the one who "opened the womb of his mother," and that was the first child. The firstborn for inheritance is the first viable child, and that is the second one.
Art: Salomon de Bray - The Twins Clara and Aelbert de Bray