Missouri's controversial eight-week abortion ban is officially being challenged in court. The complaint, filed Tuesday evening, means every one of the so-called "heartbeat" bans passed in various states is now in the hands of the courts.

In a complaint filed jointly by Planned Parenthood, the American Civil Liberties Union, and Paul Weiss LLP — the law firm that successfully fought to legalize gay marriage — attorneys argue that Missouri's ban on abortion after eight weeks is in violation of Supreme Court precedent that protects access to abortion. The state law provides no exceptions for victims or rape or incest.

"Many women don't even know that they are aware of their pregnancy, or may have a challenge getting into a clinic for eight weeks," said Alexis McGill Johnson, the newly-appointed acting president and chief executive officer of Planned Parenthood Federation of America and the Planned Parenthood Action Fund, in an exclusive interview with CBS News. "We think that is wrong, we think it is illegal and so we are fighting, you know, we are fighting."

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The Missouri law was signed in May and is set to take effect August 28.

In an unprecedented flood of anti-abortion legislation, state lawmakers introduced over 300 bills restricting access to the procedure in the first half of 2019, according to data compiled by the Guttmacher Institute, a reproductive health research organization. Six states, including Missouri, passed so-called "heartbeat" bans, laws that would prohibit abortion after a fetal heartbeat is detected in the early weeks of pregnancy. Alabama went a step further, passing a law that bans the procedure in nearly all cases.

Planned Parenthood and the ACLU have been on the front lines, bringing legal challenges to those laws. With the Missouri case, all so-called "heartbeat" bans will have been formally challenged in court. Federal judges in all states but Georgia have granted preliminary injunctions to block the laws from implementation.

In Missouri, abortion access is already difficult, even without the eight-week ban in place. Women seeking an abortion are required to undergo a 72-hour waiting period and receive state-mandated counseling designed to dissuade them from receiving an abortion. Various other regulations have closed all but one clinic in the state: a lone Planned Parenthood in St. Louis.

"If you're a person in rural Missouri, the chance that you'll be able to access abortion care is dramatically less than someone in another area," said Dr. Colleen McNicholas, chief medical officer at Planned Parenthood of the St. Louis Region, in a telephone interview with CBS News.

"For years, Missouri officials have engaged in a targeted campaign against abortion," the plaintiffs write in the complaint, which seeks an injunction blocking the legislation.

They argue that the law "will irreparably harm Plaintiffs and their patients by severely restricting access to pre-viability abortion care, preventing the vast majority of patients from obtaining the constitutionally protected medical care they seek."

In addition to the ban on abortions after eight weeks, Missouri's legislation also includes a "trigger law," which would outlaw abortion if Roe v. Wade is overturned, and a ladder of less-restrictive time limits ranging from 14 to 20 weeks, depending on what the courts later find to be constitutional. It also bans abortions based solely on race, sex or a "prenatal diagnosis, test, or screening indicating Down Syndrome or the potential of Down Syndrome." Tuesday's lawsuit challenges every provision included in the legislation.