On the day when a Brit Milah (circumcision) is conducted in the synagogue, the Tachanun prayer is omitted and Yehi Shem is recited in all the Minyanim (prayer services) held in that synagogue until the actual Brit. For example, if a synagogue hosts Minyanim at 7:00, 8:00 and 9:00 in the morning, and a Brit Milah is scheduled for after the 9:00 AM service, Yehi Shem is recited at all three Minyanim. Even if none of the Ba'alei Brit Avraham – meaning, the child's father, the Mohel (person performing the circumcision), and the Sandak (person holding the baby during the circumcision) – are present at these Minyanim, Yehi Shem is recited by virtue of the fact that a Brit Milah will take place that day in the synagogue. Furthermore, even if the circumcision will be held in the synagogue's main sanctuary and the earlier Minyanim take place in the auxiliary room, Yehi Shem will still be recited in the earlier Minyanim. See SHulchan Aruch Orach Hayim 131:4.
Likewise, a Minyan will recite Yehi Shem and omit Tachanun if one of the three Ba'alei Brit is present on the morning of the Brit, even if the Brit will not be held in that synagogue. For example, if the Brit will take place in the child's home, or in a different synagogue, Yehi Shem will nevertheless only be said if one of the three Ba'alei Brit is in attendance. (See Yabia Omaer, Helek 3, O"H, SIman 23, Ot 4)
At Mincha on the afternoon following a Brit Milah, Yehi Shem is recited and Tachanun is omitted only if one of the three Ba'alei Brit is in attendance. Otherwise, Tachanun is recited, even if a Brit was held in that synagogue earlier in the day.
In some instances, the circumcision procedure is performed by two people: the Mohel, who severs the foreskin, and the Porei'a, who uncovers the foreskin after it has been cut. (The Mohel usually performs both stages, but on some occasions a different person is summoned to perform the Peri'a – the uncovering of the foreskin.) In such instances, the Porei'a has the same status as the Mohel with respect to omitting Tachanun and reciting Yehi Shem. (See Yabia Omaer, Helek 3, O"H, SIman 12)
At Mincha on Shabbat, we generally recite Tzidkatecha after the repetition of the Shemona Esrei. Does a congregation recite Yehi Shem in its place if a father who made a Brit Milah that morning is in attendance? Or, for that matter, would Yehi Shemi be recited if a Chatan (groom within a week of his wedding) participates in the Minyan?
Chacham Ovadya Yosef ruled that Tzidkatecha is omitted during Mincha on Shabbat only if the day itself mandates omitting Tachanun, such as during the month of Nissan, on festivals, and the like. In situations of personal rejoicing, however, Tzidkatecha is not omitted. This is not the custom of our community. Hence, Yehi SHem is siad at Mincha of Shabbat if there is Baale Brit present.
1) On a day when a circumcision is held in a synagogue, all the Minyanim held in the building until the Brit Milah omit Tachanun and recite Yehi Shem.
2) If one of the three Ba'alei Brit participates in a Minyan on the day of the Brit, the Minyan omits Tachanun and recites Yehi Shem, even if the Brit will be held elsewhere.
3) At Mincha in the afternoon following a Brit, Tachanun is omitted and Yehi Shem is recited only if one of the three Ba'alei Brit is in attendance.
4) Tzidkatecha is recited at Mincha on Shabbat even if one of the Ba'alei Brit, or a Chatan, is in attendance, according to Hacham OVadia. However, the Minhag in the community is to say Yehi Shem. (See Kaf Hachayim Sofer, O"H Siman 292 Ot 18)