Many many times the Bible offers salvation to man for only 1 thing the man does: Trust Christ as Savior, Believe on the Lord Jesus Christ, Have faith in Him. Many, many times the only thing a man is to do is have faith in Christ. A few times some other expression is used; but these must be understood in the general, overwhelming Bible context. The Lord cannot lie. If he said, "Believe on the Lord Jesus Christ and you shall be saved (and He has said that), yet when you got to the Pearly Gates you were rejected, though you had believed, that would be at best false advertising, which our Lord would never do.
Once (John 1:12) the condition is receive Him, but then it is redefined as believing. One saying (repeated more than once from Joel) is "Whosoever shall call on the name of the LORD shall be saved." But this is a call of faith, as context indicates.
Romans 10: "11 For the scripture saith, Whosoever believes on him shall not be put to shame. 12 For there is no distinction between Jew and Greek: for the same Lord is Lord of all, and is rich unto all that call upon him: 13 for, Whosoever shall call upon the name of the LORD shall be saved. 14 How then shall they call on him in whom they have not believed?
One may exercise faith by calling on His name.
Repent is used, but Repent (metanoia) is primarily a change of mind, not sorrow for sin, nor promising to turn over a new leaf. The repentence (change of mind) that saves is the change from not-trusting-Christ to trusting-in-Christ (faith again).
As to water baptism:
1) The Bible never says water baptism saves,
2) baptism is not something a person does, but something done to him (he is passive).
3) inasmuch as Spirit baptism puts a man into the Body of Christ (1 Cor 12:13) and inasmuch as our incorporation into Christ puts us on the cross with Christ when He died and crucifies our Old Man (Romans 6), one can call Spirit baptism salvific -- but it is not something a man does, but is done to Him.
No, water baptism is not essential to salvation. There is but one condition of salvation in the Bible, faith. In the OT people didn't distinguish the Trinity as we do. Now we must trust in God the Son. In OT times, men trusted YHWH without distinguishing the Trinity so far as we know.
"that whosoever believes in Him should not perish, but have everlasting life."
I testify to these truths as one who has searched the Bible from cover to cover on salvation and marked all the passages in pink with a highlighter, including passages that some might use to argue against my conclusions.
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