Christ's atonement and righteousness guarantees salvation to all for whom He served as the Substitute, that God might be just

"The sin of Adam did not make the condemnation of all men merely possible; it was the ground of their actual condemnation. So the righteousness of Christ did not make the salvation of men merely possible, it secured the actual salvation of those for whom He wrought." -Charles Hodge

"If Christ has died for you, you can never be lost. God will not punish twice for one thing. If God punished Christ for your sins He will not punish you. 'Payment God’s justice cannot twice demand; first, at the bleeding Saviour’s hand, and then again at mine.' How can God be just if he punished Christ, the substitute, and then man himself afterwards?" -C.H. Spurgeon

(Both of these quotes are from Lorraine Boettner's Limited Atonement.)

These quotes regarding the atonement answer to the questions that the doctrine of justification seeks to answer: "How can a man be just with God?" and "How can God be just but yet justify the wicked?" Only personal, perfect, and perpetual obedience will suffice; we must be dressed in spotless raiment, untainted in any respect by the defilement of totally depraved flesh. In our case as sinners, in order to be judicially declared "righteous" before God, we must obtain pardon as well as acceptance through perfect, perpetual law-keeping, as well as an infinite sacrifice for our sins -- both of which are impossible for us to perform. Because Christ served as our substitute in His passive obedience -- taking the guilt, curse, and penalty for sin upon Himself through imputation -- God has justly punished sin once and for all for His elect. Christ has also purchased the eternal reward of glorified life for His elect by His active obedience, His lifetime of perfect and perpetual Law-keeping from start to finish -- fulfilling all righteousness -- that is imputed to us. The question now becomes: "How can God be just if He punishes a man, for whom Christ died, with an eternity of hell apart from glorified life with God?"

Christ died that God might be just and the justifier of those who believe in Jesus (Rom. 3:26). Those who, by the instrumentality of faith alone, receive and rest in the person and work of Christ alone -- His active and passive obedience -- are pardoned and accepted, declared righteous before God's throne, and will obtain the eternal reward of glorified life that Christ purchased. God would be unjust to twice punish sin, first in the person of Christ through imputation of man's guilt when Christ bore their sins upon the tree, and secondly in the punishment of the man himself for the same sin. Arminianism and all other deviant forms of soteriology make God out to be an unjust judge. Either God is supposed to pardon and accept men on grounds other than an infinite sacrifice to pay the penalty for sins, as well as perfect, perpetual, and personal Law-keeping in every word, thought, deed, and motive; or He is supposed to exact the penalty for sin both from Christ and from some of those for whom He was supposed to die.

Lorraine Boettner: Arminianism limits the atonement more severely than does Calvinism

"When the atonement is made universal its Inherent value is destroyed. If it is applied to all men, and if some are lost, the conclusion is that it makes salvation objectively possible for all but that it does not actually save anybody. According to the Arminian theory the atonement has simply made it possible for all men to co-operate with divine grace and thus save themselves — if they will. But tell us of one cured of disease and yet dying of cancer, and the story will be equally luminous with that of one eased of sin and yet perishing through unbelief. The nature of the atonement settles its extent. If it merely made salvation possible, it applied to all men. If it effectively secured salvation, it had reference only to the elect. As Dr. Warfield says, 'The things we have to choose between are an atonement of high value, or an atonement of wide extension. The two cannot go together.' The work of Christ can be universalized only by evaporating its substance.

"Let there be no misunderstanding at this point. The Arminian limits the atonement as certainly as does the Calvinist. The Calvinist limits the extent of it in that he says it does not apply to all persons (although as has already been shown, he believes that it is efficacious for the salvation of the large proportion of the human race); while the Arminian limits the power of it, for he says that in itself it does not actually save anybody. The Calvinist limits it quantitatively, but not qualitatively; the Arminian limits it qualitatively, but not quantitatively. For the Calvinist it is like a narrow bridge which goes all the way across the stream; for the Arminian it is like a great wide bridge which goes only half-way across. As a matter of fact, the Arminian places more severe limitations on the work of Christ than does the Calvinist." (Lorraine Boettner, Limited Atonement)

Lorraine Boettner: Three Essential Imputations in Christian Theology

Lorraine Boettner on the biblical doctrine of imputation:

In Christian theology there are three separate and distinct acts of imputation. In the first place Adam's sin is imputed to all of us, his children, that is, judicially set to our account so that we are held responsible for it and suffer the consequence of it. This is commonly known as the doctrine of Original Sin. In the second place, and in precisely the same manner, our sin is imputed to Christ so that He suffers the consequences of it. And in the third place Christ's righteousness is imputed to us and secures for us entrance into heaven. We are, of course, no more personally guilty of Adam's sin than Christ is personally guilty of ours, or than we are personally meritorious because of His righteousness. In each case it is a judicial transaction. We receive salvation from Christ in precisely the same way that we receive condemnation and ruin from Adam. In each case the result follows because of the close and official union which exists between the persons involved. To reject any one of these three steps is to reject an essential part of the Christian system. (Lorraine Boettner, The Atonement; quoted in Rushdoony's Systematic Theology, p. 593)