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Mormonism and priesthood/Christians don't need a mediating priesthood
Some Christians claim that we do not need a mediating priesthood since it has been "fulfilled in Christ."
Question: Do we do not need a mediating priesthood since it has been "fulfilled in Christ?"
Some sectarian Protestants claim that Christians do not need a mediating priesthood. Is a priesthood not needed since it has been "fulfilled in Christ?"
The Bible repeatedly speaks of a priesthood authority outside of Jesus both before and after His resurrection, with John describing such callings just prior to the second coming.
Early Christian authors insisted too that high priests, prophets, bishops, elders, priests, and deacons with authority persisted among the Christians.
Efforts to deny the need for a formal priesthood seem to arise mostly out of theological necessity, rather than historical or biblical evidence.
This criticism usually comes from Protestant circles and usually involves an argument for some form of "the priesthood of all believers."
The critics' theological need to dispense with priesthood authority—since Protestantism cannot claim authority from either a restoration (which they deny) or a continuation (having broken with Catholicism)—leads them to ignore the clear evidence from the early Church.
As William Hamblin pointed out:
Why, if Christ has removed all need for human priesthood authority, did Christ order the lepers he healed to go to the Jewish priests for purification (Mark 1:44, Luke 17:14)? Apparently Christ believed that his miraculous powers of healing did not negate or supercede the priesthood authority of the Jewish priests. Possibly Protestants could argue that Christ had not yet ascended into heaven and replaced the Jewish High Priest. If so, why does Peter speak of a post ascension "holy priesthood" (1 Peter 2:5) and "royal priesthood" (1 Peter 2:9) among Christians? Likewise John in Revelation speaks of the saints as "priests to his [Christ's] God and Father" (Revelation 1:6), and "priests to our God" (Revelation 5:10); in the resurrection there "shall be priests of God and of Christ" (Revelation 20:6). What odd statements for an infallible book to make if [the critics'] understanding of priesthood is correct.
Some of the earliest Christians also explicitly disagree with White's claims. The author of the Didache, (one of the earliest post-New Testament Christian documents, late first to early second century), states explicitly that "the prophets ... are your high priests" (13.1). Note the plural here: the prophets (profetais) are the Christians' high priests (archiereis). So, early post-New Testament Christians had prophets (a thing [James] White believes Christians shouldn't have) who were high priests (a thing [James] White believes Christians shouldn't have); and there were apparently simultaneously more than one high priest. 
Evidence after the New Testament
Wrote another author:
While there are few New Testament references to priests, other than Jesus Christ and converted Levite priests (Acts 6:7), Protestants should not assume that this office was abolished. The early church had priests along with bishops and deacons. Origen (ca. 240 A.D.) spoke of the church hierarchy in the 2nd century describing the priest's office as being between that of the deacon and bishop (Jean Danielou, "Origen", p.44-45, 49-50; Cel. 5,3,1; De Princ. 3,2,4; Hom. Luc., 35; Hom. Ez. 1,7) and Eusebius (ca. 300 A.D.) clearly distinguished between those holding the priesthood (i.e. bishops, presbyters or elders, priests, deacons, etc.) and the lay members both men and women. (Eusebius, History of the Church, 6:19, 23, 43; 7:30; 10:3, 4) Eugene Seaich observes that "documents from the early Church show that the Aaronic Priesthood did not immediately disappear from Christianity. 1 Clement (ca. 96 A.D.) divides the priesthood into High Priests, Priests and Levites. The latter were also called "Deacons" and according to Justin's First Apology (ca. 150 A.D.) were responsible for passing the bread and wine to those attending service" (Ancient Texts and Mormonism, p. 59). Though the title priest was rarely used in the New Testament, so also were similar priesthood titles such as pastor (Ephesians 4:11), evangelist (Acts 21:8); (2 Timothy 4:5), presbytery (1 Timothy 4:14), and seventy (Luke 10:1),(Luke 10:17). 
(See also 1 Clement 40-44; an article in Ferguson's Encyclopedia of Early Christianity, pp. 754-5 provides references to other second and early third century Christian sources mentioning Christian priests.)
To see citations to the critical sources for these claims, click here