The Intention(s) of Christ in the Atonement
Christ died for the elect in a way that he did not for the non-elect. It could be argued that there are at least two intentions in Christ's work of the atonement. The first one is for the elect, in whom God has decreed that the atonement is certain and eventual, therefore it is necessary that it will be applied on His people at a time of the Spirit's choosing. The second is for the non-elect, who receive non-redemptive benefits. These may include not immediately receiving (a putting off of) God's just wrath, but the redemptive benefits of the atonement will only be heralded in the divine command but never actualized in the non-elect. This is why the Bible proclaims that we freely hold out the gospel to all unbelievers and tell them that Christ died, not for all men, but for all sinners who would believe (John 3:16). The redemptive benefits are only for believers. All who believe will have the benefits of the atonement applied to them and be justified, but since no one naturally fits this description, the only persons who come to Christ are those who are God's elect, regenerated by the Holy Spirit unto faith. This means that we are wholly dependent on the work of Christ for our redemption, which includes our ability to have faith in the Redeemer. Since we do not have the power in ourselves to do anything apart from the work of God's Son on the cross, He also gives, as one of its benefits, the Holy Spirit for our conversion (1 Pet 1:3). God "has blessed us with every spiritual blessing in the heavenly places in Christ" (Eph. 1:3). Our redemption in Christ is the wellspring out of which flows regeneration, faith, repentance, justification and sanctification. So although the atonement may have more than one intent, its central purpose is for the redemption of elect (Titus 2:14). In other words, Christ died in a way for the elect that he did not for the non-elect.
Those who say they believe the doctrines of grace, but not particular redemption, carve out the Person, heart and source of grace in these doctrines. For without the particularly of the death of Jesus Christ, all of the other doctrines of grace become mere impersonal abstractions. Jesus Christ must be at their center or not at all. What do I mean by this? For example, if one claims the the atonement is not particular, but that irresistible grace is, then where does the grace come from which makes irresistible grace particular? From Christ or from some generic grace apart from Christ? If from Christ, then you already affirm particular redemption and the impersonal abstraction of four-point calvinism is overthrown. If not from Christ, do you then believe there are redemptive benefits to be had apart from Christ? It is impossible, in other words, to believe in irresistible grace without acknowledging that all spiritual/redemptive blessings (including irresistible grace) flow from one source: Christ.
To show that this doctrine has biblical authority have a look at the following: The Jewish high priest in the Old Testament prayed for the Israelites on the Day of Atonement. He approached the Lord wearing the names of the twelve tribes when bringing the sacrifice. Likewise, before His death on the cross, Jesus also prays a "high priestly prayer" for His people in John 17. Notice that in verse 9, Jesus (as high priest) prays an exclusive prayer for certain people and not others: 'I ask on their behalf; I do not ask on behalf of the world, but of those whom You have given Me; for they are Yours." Then soon after in verse 19 when praying for the same persons the Father has "given" Jesus, He prays: "For their sakes I sanctify Myself, that they themselves also may be sanctified in truth." So Jesus only sanctifies Himself for those the Father has given him, meaning these are the very ones He dies redemptively for. In short, Jesus prays for the same persons He dies redemptively for. Only those God has elected in Christ from eternity are given the grace of regeneration that they might believe. And this regeneration is among the redemptive benefits of Christ given to the elect alone - for he purchased with his blood people OUT OF every tribe, nation, tongue and people (Rev 5:9).