Where you are going wrong is that you are seeing regeneration as identical to justification. Very briefly (I don't have much time):-
The Lord opens our heart,
We repent and trust in Christ,
We are justified by God.

We are not 'saved' until we are justified, hence all those texts you have found saying that salvation is by faith. We all know that. But no one believes for salvation until the Lord gives him life (John 6:44).

I do assure you that dead men can't believe. Listen; Lazarus is dead, right? Now it's no use you or me telling Lazarus how bad it is to be dead, and the serious consequences that flow from being dead. Nor is it any use telling him how much better it would be if he were alive. He can't hear us because he's dead. And actually, it's even worse than that; Lazarus, quite frankly, is a stinker (John 11:39, KJV). He's not just dead, he's dead in trespasses and sins; he has no ability in himself, nor any right to come to life.

What hope, then, for Lazarus? None in himself, he's dead; and none in us, we can't bring him to life. But when the Son of God calls, 'Lazarus, come forth!', then, and only then will he come to life (cf. John 5:21), then he will believe on Christ and be saved.

You can see this very clearly in Acts 16. First Lydia has her heart opened by the Lord, then she believes. The same is true of the Philippian jailor. First the Lord opens his heart giving him spiritual life, then, 'He fell down trembling before Paul and Silas.....and said, "Sirs, what must I do to be saved?"'(Acts 16:29-30). Do you see? Although he was under conviction, he wasn't yet saved, so Paul says, 'Believe on the Lord Jesus Christ and you will be saved.' He did and he was (v34). First regeneration, then repentance and faith; the justification and eternal life.

I hope that's helpful; sorry it's a bit short.

Every blessing,

Itinerant Preacher & Bible Teacher in Merrie England.
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