Many Christians argue that besides Melchizedek, only Christ is a holder of this priesthood. They often refer to Hebrews which gives the impression that they may be correct. Is there any chance to prove the LDS version of this priesthood just from the bible alone ?
The Bible does not refer to others holding the Melchizedek Priesthood, other than Melchizedek and Christ himself. It does refer to offices that are part of the Melchizedek Priesthood. Prophets, Apostles, Seventies, Elders and High Priests are all offices of the Melchizedek Priesthood and these offices of the Priesthood are all mentioned in the Bible.
It is through modern scriptures that we are taught that these offices belong to the higher or Melchizedek Priesthood. In the Doctrine and Covenants section 84 verses 6 through 16, we are told how the Melchizedek Priesthood was passed from Adam to Moses. In the days of Moses, the Lord became angry with the children of Israel and He took the Melchizedek Priesthood and Moses out of their midst. D&C 84: “24 But they hardened their hearts and could not endure his presence; therefore, the Lord in his wrath, for his anger was kindled against them, swore that they should not enter into his rest while in the wilderness, which rest is the fulness of his glory. 25 Therefore, he took Moses out of their midst, and the Holy Priesthood also.”
Aaron held the Melchizedek Priesthood (D&C 84: 18), but was unable to confer this Priesthood on others. The prophets between Moses and Jesus would have also held the Melchizedek Priesthood, but would have had this priesthood conferred on them directly as did Joseph Smith and Oliver Cowdery.
In Matthew chapter 16, Peter is told by Jesus that he will be given the keys of the Kingdom of Heaven: “19 And I will give unto thee the keys of the kingdom of heaven: and whatsoever thou shalt bind on earth shall be bound in heaven: and whatsoever thou shalt loose on earth shall be loosed in heaven.” Peter received these keys on the Mount of Transfiguration from Elijah. Elijah, who held these keys, also restored these to the Prophet Joseph Smith in the Kirtland Temple. Only one holding the Melchizedek Priesthood would be able to hold these keys as does President Monson today.