What is the wisdom behind legislating the Udhiyah?
The Muslim must know that through the Uḍḥiyah, he complies with the command of Allah, the Exalted, and performs a worship that brings him closer to Allah and distances him from the Fire.
The Uḍḥiyah holds profound wisdoms and noble meanings, including:
1-Reviving the Sunnah of Abraham, peace be upon him: He complied with Allah's command when He ordered him to slaughter his son Ishmael. Abraham, peace be upon him, succeeded in the test, and when he was about to slaughter his son, Allah sent down a ram to ransom Ishmael and commanded him to slaughter it instead. Allah, the Exalted, says {what means]: "And when he reached with him [the age of] exertion, he said, 'O my son, indeed I have seen in a dream that I [must] sacrifice you, so see what you think.' He said, 'O my father, do as you are commanded. You will find me, if Allah wills, of the steadfast.' And when they had both submitted and he put him down upon his forehead, We called to him, 'O Abraham, You have fulfilled the vision.' Indeed, We thus reward the doers of good. Indeed, this was the clear trial. And We ransomed him with a great sacrifice." [Al-Ṣāffāt/102-107]. Thus, the sacrificial offering became an enduring Sunnah until the Day of Judgment, and Muslims offer their wealth as sacrifices, drawing closer to Allah, the Exalted, reviving this great Sunnah.
2-Providing abundance for people on Eid day and the Days of Tashrīq: Among the wisdoms of the Uḍḥiyah is that the Muslim provides generously for his family, neighbors, relatives, and the poor during these days. The Muslim is encouraged to eat from his sacrifice, give charity from it to the poor, and gift from it to his wealthy neighbors. Thus, goodness spreads throughout the entire community. The Prophet, peace and blessings be upon him, said: "The Days of Tashrīq are days of eating and drinking." (Narrated by Muslim). Allah, the Exalted, says {what means}: "And the camels and cattle We have appointed for you as among the symbols of Allah; for you therein is good." [Al-Ḥajj/36]. This good includes the good of both this world and the Hereafter.
Is it permissible for one who sacrifices on behalf of another, with the latter's permission, to eat from the sacrifice?
It is permissible for someone who sacrificed on behalf of another with their permission to eat from it with their permission, and they stand in their place (act as their representative) in distributing it.
It is stated in Nihayat al-Muhtaj ila Sharh al-Minhaj (Vol.8/P.141): "And he—meaning the one sacrificing on his own behalf, provided he has not apostatized—has the right to eat from a voluntary sacrifice and his sacrificial gift; rather, it is recommended. As for an obligatory sacrifice, it is forbidden for him to eat from it, whether it was designated as such initially or as a liability in his dhimmah. And excluded by what has preceded is if he sacrifices on behalf of another, or if he apostatized, then it is not permissible for him to eat from it, just as it is absolutely impermissible to feed a disbeliever from it."(The text I found indicates the impermissibility of eating) ???
And it is stated in Hashiyat al-Jamal ‘ala Sharh al-Manhaj (Vol.5/P.262): "If a person offers a sacrifice on behalf of a living person with their permission, does he act as their representative in distributing it—on the grounds that permission to sacrifice is permission to distribute it—or does it depend on their explicit permission? This requires consideration, but the first view is not far-fetched." And Allah the Almighty knows best.
Is vomit among the nullifiers of Wudu (ablution)?
Vomit does not nullify Wudu, but it is a Najaasah (impurity) that requires rinsing the mouth and washing whatever became dirty by it since the prayer becomes valid only after the removal of Najaasah. And Allah Knows Best.
What is the Sharia basis of the Udhiyah?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
The Sharia basis of the Udhiyah (sacrificial offering) is firmly established through the Quran, the Sunnah, and the Consensus (Ijma') of the Muslims:
1. Evidence from the Holy Quran
Allah the Exalted says {what means}: "And the camels and cattle We have appointed for you as among the symbols of Allah; for you therein is good." [Al-Hajj/36]. He also says {what means}: "So pray to your Lord and sacrifice [to Him alone]" [Al-Kawthar/2]. According to the most well-known scholarly interpretations of this verse, "prayer" refers to the Eid prayer, and "sacrifice" refers to the slaughtering of the Udhiyah.
2. Evidence from the Sunnah
Al-Bara' bin 'Azib (may Allah be pleased with him) narrated that the Prophet ﷺ said: "The first thing we start with on this day of ours is to pray, then we return and sacrifice. Whoever does that has followed our Sunnah, and whoever slaughters before [the prayer], it is merely meat he has provided for his family; it is not part of the ritual sacrifice in any way" [Reported by Bukhari & Muslim].
Anas (may Allah be pleased with him) narrated: "The Prophet ﷺ sacrificed two white rams with horns. He slaughtered them with his own hand, mentioned the name of Allah (Tasmiyah), and said the Takbir" [Reported by Bukhari & Muslim].
3. Evidence from Scholarly Consensus (Ijma')
The Muslims have reached a unanimous consensus on the Sharia basis of the Udhiyah, and no one among the scholars has disagreed with this. [Al-Sherbini, Mughni al-Muhtaj, Vol.6/P.122].And Allah the Exalted knows best.
