How the Trinity Works Together in Salvation
One of the most important insights of Reformed theology is the unity of the works of the Trinity. Calvinists believe that God the Father, God the Son, and God the Holy Spirit are united in the work of redeeming lost mankind. We do not believe that they act against one another or even on one another, but with one another in our salvation. For instance, Jesus did not die to convince the Father to change His attitude toward us from enmity to love. Rather, Jesus died on the cross because of the Father's love for us, as John 3:16 says: "For God so loved the world that he gave his only Son, that whoever believes in him should not perish but have eternal life." The Father and Son are united in their work for the salvation of those who believe: the Father electing and sending His Son; the Son atoning for the sins of those chosen and given to Him by the Father (John 6:37–40). The same harmony exists between the Son and the Spirit. Jesus did not die for the sins of all people, only to have the Holy Spirit apply the benefits of His work merely to some. Rather, the Holy Spirit regenerates precisely the people for whom Jesus offered His atoning death, so that the work of the second and third persons of the Trinity harmonizes perfectly.
The Holy Spirit regenerates precisely the people for whom Jesus offered His atoning death
This emphasis on the unity of the Trinity in salvation may be seen in the doctrines of grace as organized by the TULIP acrostic. The doctrines of grace start with a problem: T—man's total depravity. The answer to this problem is brought about by the unified work of the Trinity. It begins with unconditional election, which is focused on the Father's sovereign purpose in predestination. Salvation is then accomplished by the atoning work of Jesus on the cross, which, according to limited atonement, was offered for those elected by the Father. This salvation is then sovereignly applied to those same elect individuals by the regenerating work of the Holy Spirit, which is the point of irresistible grace. Therefore, just as unconditional election describes the grace of the Father and limited atonement describes the grace of the Son, irresistible grace presents the grace of the Holy Spirit. While acknowledging that where one member of the Trinity acts, all are involved, we may identify irresistible grace as the Holy Spirit's own doctrine of grace.
An excerpt from What's So Great about the Doctrines of Grace? by Richard Phillips.