In a sense, the sacrifice changed from the offering to the offerer.

 One thing remained the same in all of these offerings: everything about Mosaic sacrifice focused on Christ....

“Therefore, it is expedient that there should be a great and last sacrifice, and then shall there be … a stop to the shedding of blood; then shall the law of Moses be fulfilled. …

“And behold, this is the whole meaning of the law, every whit pointing to that great and last sacrifice; and that great and last sacrifice will be the Son of God, yea, infinite and eternal” (Alma 34:13–14).

Now, here is a very important truth: We should understand that the law of Moses is not the same thing as the law of sacrifice. While the law of Moses was fulfilled, the principles of the law of sacrifice continue to be part of the doctrine of the Church. The primary purpose of the law of sacrifice still is to test us and assist us in coming unto Christ. After the Savior’s ultimate sacrifice, two adjustments were made in the practice of this law. First, the ordinance of the sacrament replaced the ordinance of sacrifice; and second, this change moved the focus of the sacrifice from a person’s animal to the person himself. In a sense, the sacrifice changed from the offering to the offerer.

M. Russell Ballard, The Law of Sacrifice, October 1998 General Conference