It will argue the current legislation on cases of rape, incest or fatal foetal abnormality is in breach of the European Convention on Human Rights (ECHR).
Northern Ireland's abortion law is much stricter than in the rest of the United Kingdom where the 1967 abortion act applies. The government should not overstretch its reach - it does not have a mandate to intervene in Northern Ireland; abortion is a devolved issue and should be decided by the Northern Ireland Assembly.
The law is far less restrictive in the rest of the United Kingdom, and hundreds of Northern Irish women travel to England every year to have unwanted pregnancies terminated.
Over the next three days, judges at the UK Supreme Court will hear an appeal by the NIHRC and other pro-abortion groups, including Amnesty International, claiming that the abortion of severely disabled babies is a human right.
Anyone who unlawfully carries out an abortion not under these circumstances could be jailed for life.
The plan will ensure that abortion is free for women travelling from Northern Ireland at the point of delivery, rather than being paid for beforehand and then being reimbursed.
Christian campaigners have reacted with disappointment after the United Kingdom government revealed plans to provide free abortion services in England for women from Northern Ireland, where terminations are heavily restricted.
The appeal will explore whether abortion law here is incompatible with global human rights.
Mr Allamby added: "This case is of great significance in the United Kingdom and internationally, just four months after our appeal the Supreme Court has granted an expedited hearing".
The Northern Ireland Assembly voted in February past year against legalising abortion in cases of fatal foetal abnormality and rape or incest. "It's an act of lethal violence directed at an unborn child".
It will also rule on whether the Northern Ireland Act 1998 entitles the NIHRC to bring the proceedings under the Human Rights Act 1998 to seek a declaration of incompatibility.
The Northern Irish Department of Justice and the Attorney General successfully appealed this ruling.
In written argument before the court, Ms Lieven said both the Attorney General and the Department for Justice "are at pains to suggest there is no jurisdiction" for the Supreme Court to consider the challenge, because of the commission's "lack of standing and a failure to identify an unlawful act".
Campaigners gathered outside the court and say they will remain as the appeal continues.
"It is time for the Supreme Court to step in and do what our government has failed to do - protect the long-neglected human rights of women and girls in a part of the United Kingdom", she said.