The case of Ashers rose to light after an order for a cake with a message of support for gay marriage was refused, as it went against the owners' Christian principles.
Last month at Belfast County Court, District Judge Isobel Brownlie said they'd acted unlawfully and ordered them to pay £500 damages, after stating the customer had been treated "less favourably".
Lawyers for the bakery, run by the McArthur family, say they will fight the case on the idea that the law was misinterpreted.
In papers lodged this week they said: "We have identified the broad points of law with which the applicant is dissatisfied."
It means the case won't be heard again and witnesses won't be made to give evidence again.
Speaking last week about the decision to appeal, the McArthur family said: "After much careful and prayerful consideration given to legal advice, we have decided to appeal the judgment handed down last Tuesday.
"We continue to insist that we have done nothing wrong as we have discriminated against no individual but rather acted according to what the Bible teaches regarding marriage.
"As many other people have already noted, Christian beliefs seem to have been trampled over in this judgment and we believe this only has negative effects for our society.
Their case last week received support from an unusual source. The actor Patrick Stewart, who has campaigned for gay marriage, said Ashers had been treated unfairly and that he was on their side.
There are six main grounds for the appeal:
- Whether the Judge was correct as a matter of law to hold that Ashers had discriminated against Mr Lee directly on grounds of sexual orientation?
- Whether the Judge was correct as a matter of law to hold that Ashers had discriminated against Mr Lee directly on grounds of religious belief or political opinion?
- Whether the Judge was correct as a matter of law to hold that, had she not considered the case to give rise to direct discrimination on any of the above protected characteristics, alternatively she would have held that the same amounted to indirect discrimination which was not justified by Ashers?
- Whether the Judge was correct to hold as a matter of law that it was not necessary to read down or disapply the provisions of the Regulations or FETO (The Fair Employment and Treatment (NI) Order 1998) to take account of Ashers' protected right to hold and manifest their genuinely-held religious belief that marriage is, according to God's law, between one man and one woman, pursuant to Article 9 of the European Convention on Human Rights (ECHR).
- Whether the Judge was correct to hold as a matter of law that it was not necessary to read down or disapply the provisions of the Regulations or FETO to take account of Ashers' protected right not to express a religious belief or political opinion which was contrary to their conscience, contrary to Article 10 of the ECHR.
- Whether the Judge was correct to hold as a matter of law that Ashers was not entitled to protection as a result of the rights under Articles 9 and/or 10 of the ECHR.