Commentary for Kiddushin 69:11
ואי אשמועינן הא משום דחיותה היא אבל כופר אימא
For if the first [only] were stated, [l would say] that the All-Merciful had compassion upon her [woman], for the sake of atonement;<span class="x" onmousemove="('comment',' The first refers to sacrifice for sin, and the woman is given the same opportunity of atoning as man.');"><sup>21</sup></span> but as for civil law, I might argue that it applie only to man, who engages in commerce, but not to woman, who does not. While if the second [alone] were iintimated, that is because oneðs livelihood depends thereon;<span class="x" onmousemove="('comment',' Viz., on the protection afforded by civil law.');"><sup>22</sup></span> but as for ransom ,<span class="x" onmousemove="('comment',' The last law quoted treats of the ransom paid by the owner of the ox; vv.29H');"><sup>23</sup></span> I might argue,
Daf Shevui to Kiddushin
The first would not be sufficient to teach that women are subject to all civil law because women did not engage in commerce as did men.
The second, about civil law, would not be sufficient to teach that if an animal kills a woman the owner must pay a ransom because women are not obligated in all commandments, as are men. (I know, this is a bit of an ugly one. But I do think it is a faithful representation of what happens when you exempt people from commandments. You might start to think that there life is less worthy than those fully commanded. This seems to be what might have happened to at least one strand of rabbinic thought).
If all we had was the third rule we might have thought that in that case, men and women were equated because it is a matter of life. But in cases of atonement or civil laws, they are not equal. Therefore we need all three derivations to teach that they are equal in all negative commandments.