Melacha on day of Milah:[1]

Some Poskim[2] rule that the father of the newborn is forbidden to perform Melacha [i.e. haircut, bathing, etc[3]] on the day circumcision until the circumcision takes place, as is the law regarding any other time-dependent Mitzvah. However, the mother of the newborn is permitted to perform Melacha as normal prior to the circumcision.[4] Furthermore, some Poskim[5] rule that the above prohibition even to the father of the newborn only applies if he himself will be doing the circumcision however, if he has appointed a Mohel to do so, then it is permitted for him to perform Melacha as usual prior to the circumcision. However, in such a case, some Poskim[6] rule that the Mohel is prohibited from doing Melacha prior to the circumcision unless the circumcision is not taking place on time. Other Poskim[7] argue on all the above and rule that there was never a prohibition decreed against doing Melacha prior to doing the circumcision even for the father the child, and even if he himself is the Mohel.

Sandek: According to all, a Sandek may perform Melacha on the day of the Bris.

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[1] See Pesakim Uteshuvos 262:6 footnotes 44-45

[2] See Nachlas Tzevi 262:1; Daas Torah of Maharsham 262; Peri Hasadeh 3:75; Poskim in Pesakim Uteshuvos 262:6 footnote 39 and 44

[3] See Michaber 232:2

[4] Nachlas Tzevi ibid

[5] Nachlas Tzevi ibid; See Poskim in Pesakim Uteshuvos 262:6 footnote 40 for dissenting opinions

[6] Nachlas Tzevi ibid; See Pesakim Uteshuvos 262:6 footnote 40 in name of Betzel Hachochma 4:60 for dissenting opinions

[7] Maharam Shick O.C. 287; Yad Shaul 265; Pesakim Uteshuvos 262:6 footnote 41