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2 add rav ovadia
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For further discussion about whether a Beit Din may sign off/administer the wedding, see the Machloket between Rav Moshe Feinstein (Iggerot Moshe Even HaEzer vol 1 Siman 5), who forbade it, and Rav Shalom Meshash (Shemesh U’Magen vol. 3 Even HaEzer 58) and Rav Shlomo Amar (Shema Shlomo vol. 5 Even HaEzer 8), who permitted it, as discussed in by Rav J.D. Bleich, Rav Elyashiv Knohl, a Beit Din in Haifa, and Rav Howard Jachter. Rav Ovadia in later years was mattir outright based on a double Safek: whether he's really a kohen or not and if the Halacha follows the Rishonim and Shulchan Aruch who hold a Bat Nochri is pegumah or like those who disagree. (Yabia Omer vol. 11 Even HaEzer 34)

Regarding the children's status, see the links, as that is also up for debate. The Beit Din in Haifa quotes a previous ruling negating their status and then argues in favor of them being Kohanim.

Pasting from https://judaism.stackexchange.com/a/87149/13871

For further discussion about whether a Beit Din may sign off/administer the wedding, see the Machloket between Rav Moshe Feinstein (Iggerot Moshe Even HaEzer vol 1 Siman 5), who forbade it, and Rav Shalom Meshash (Shemesh U’Magen vol. 3 Even HaEzer 58) and Rav Shlomo Amar (Shema Shlomo vol. 5 Even HaEzer 8), who permitted it, as discussed in by Rav J.D. Bleich, Rav Elyashiv Knohl, a Beit Din in Haifa, and Rav Howard Jachter.

Regarding the children's status, see the links, as that is also up for debate. The Beit Din in Haifa quotes a previous ruling negating their status and then argues in favor of them being Kohanim.

Pasting from https://judaism.stackexchange.com/a/87149/13871

For further discussion about whether a Beit Din may sign off/administer the wedding, see the Machloket between Rav Moshe Feinstein (Iggerot Moshe Even HaEzer vol 1 Siman 5), who forbade it, and Rav Shalom Meshash (Shemesh U’Magen vol. 3 Even HaEzer 58) and Rav Shlomo Amar (Shema Shlomo vol. 5 Even HaEzer 8), who permitted it, as discussed in by Rav J.D. Bleich, Rav Elyashiv Knohl, a Beit Din in Haifa, and Rav Howard Jachter. Rav Ovadia in later years was mattir outright based on a double Safek: whether he's really a kohen or not and if the Halacha follows the Rishonim and Shulchan Aruch who hold a Bat Nochri is pegumah or like those who disagree. (Yabia Omer vol. 11 Even HaEzer 34)

Regarding the children's status, see the links, as that is also up for debate. The Beit Din in Haifa quotes a previous ruling negating their status and then argues in favor of them being Kohanim.

1
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Pasting from https://judaism.stackexchange.com/a/87149/13871

For further discussion about whether a Beit Din may sign off/administer the wedding, see the Machloket between Rav Moshe Feinstein (Iggerot Moshe Even HaEzer vol 1 Siman 5), who forbade it, and Rav Shalom Meshash (Shemesh U’Magen vol. 3 Even HaEzer 58) and Rav Shlomo Amar (Shema Shlomo vol. 5 Even HaEzer 8), who permitted it, as discussed in by Rav J.D. Bleich, Rav Elyashiv Knohl, a Beit Din in Haifa, and Rav Howard Jachter.

Regarding the children's status, see the links, as that is also up for debate. The Beit Din in Haifa quotes a previous ruling negating their status and then argues in favor of them being Kohanim.