how did the antislavery movement help spur the women's rights movement?

The answer to this question is not as clear as some might think. The anti-slavery movement (also known as abolition) was primarily the work of New Englanders. Since women were widely excluded from participating in politics and civic life in general (including the church) the aboltion movement was something were women could flex thier political muscle. It was during the abolition movement that women's organizations first started to appear in large numbers. Thus after the war (Civil) women felt empowered by their ability to change things (this is not to say that women won the war, but the greater point that they contributed to the aggitation against slavery). See the Seneca Falls convention for more information.

One idea too is, when one group is denied their civil rights, for any reason, then it's easier to deny other groups their civil rights. The opposite is also true, when others have their rights, then those rights can easily be spread to other groups. Thats why the ACLU fights soo hard to protect the rights of groups, even ones most people despise. If you can prevent the KKK or the Nazi's from marching, then it's easier to stop blacks or hispanics from marching. If you allow christian prayer in public school, then you also have to allow muslim, budhist and hebrew prayers to be performed. Slippery Slope.