Well that is an error as even Calvinist theologians confess
Faith as an Instrumental cause
John 3:16 (KJV 1900) — 16 For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.
The efficient (primary) cause of our eternal salvation the Scripture uniformly proclaims to be the mercy and free love of the heavenly Father towards us; the material cause to be Christ, with the obedience by which he purchased righteousness for us; and what can the formal or instrumental cause be but faith? John includes the three in one sentence when he says, “God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish but have everlasting life,” (John 3:16).
That is not the word of an Arminian or a provisionist but of
John Calvin, Institutes of the Christian Religion (Bellingham, WA: Logos Bible Software, 1997).
Similarly The Calvinist Theologian Berkouwer states
3. TO FAITH. Faith is the mediate or instrumental cause of sanctification as well as of justification. It does not merit sanctification any more than it does justification, but it unites us to Christ and keeps us in touch with Him as the Head of the new humanity, who is the source of the new life within us, and also of our progressive sanctification, through the operation of the Holy Spirit
L. Berkhof, Systematic Theology (Grand Rapids, MI: Wm. B. Eerdmans publishing co., 1938), 537.
There is not just a correlation between faith and justification, faith and regeneration , faith and salvation
There is a causal relationship even Calvinist theologians have affirmed
see also
It may be asked: If atonement naturally and necessarily cancels guilt, why does not the vicarious atonement of Christ save all men indiscriminately, as the universalist contends? The substituted suffering of Christ being infinite is equal in value to the personal suffering of all mankind; why then are not all men upon the same footing and in the class of the saved, by virtue of it? The answer is because it is a natural impossibility. Vicarious atonement without faith in it is powerless to save. It is not the making of this atonement, but the trusting in it, that saves the sinner: “By faith are you saved” (Eph. 2:8); “he that believes shall be saved” (Mark 16:16). The making of this atonement merely satisfies the legal claims, and this is all that it does. If it were made but never imputed and appropriated, it would result in no salvation. A substituted satisfaction of justice without an act of trust in it would be useless to sinners. It is as naturally impossible that Christ’s death should save from punishment one who does not confide in it as that a loaf of bread should save from starvation a man who does not eat it. The assertion that because the atonement of Christ is sufficient for all men therefore no men are lost is as absurd as the assertion that because the grain produced in the year 1880 was sufficient to support the life of all men on the globe therefore no men died of starvation during that year. The mere fact that Jesus Christ made satisfaction for human sin, alone and of itself, will save no soul. Christ, conceivably, might have died precisely as he did and his death have been just as valuable for expiatory purposes as it is, but if his death had not been followed with the work of the Holy Spirit and the act of faith on the part of individual men, he would have died in vain.
[1]
[1] William Greenough Thayer Shedd,
Dogmatic Theology, ed. Alan W. Gomes, 3rd ed. (Phillipsburg, NJ: P & R Pub., 2003), 726.