The sacrifice changed from the offering to the offerer

“After the Savior’s ultimate sacrifice,  the ordinance of the sacrament replaced the ordinance of sacrifice.  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. 

“ Instead of the Lord requiring our animals or grain, now He wants us to give up all that is ungodly.  Elder Neal A. Maxwell of the Quorum of the Twelve Apostles said: ‘Real, personal sacrifice never was placing an animal on the altar. Instead, it is a willingness to put the animal in us upon the altar and letting it be consumed!’ (‘Deny Yourselves of All Ungodliness,’ Ensign, May 1995)

“When we overcome our own selfish desires and put God first in our lives and covenant to serve Him regardless of the cost, we are then living the law of sacrifice” (“The Law of Sacrifice,” Ensign, Oct. 1998, 10).