The "born-again before faith" doctrine you espouse is utterly false and a blatant creation of the Calvinist apologist to set aside the fact that fallen men can and do believe in the Gospel message. The order the Bible teaches is the opposite, as usual. Men believe, AND THEN they are regenerated. Prior to this, the Holy Spirit CONVICTS THEM of their sin, He does not regenerate them until AFTER they believe-Eph 1:12-13:

12 That we should be to the praise of his glory, who first trusted in Christ.

13 In whom ye also [trusted], after that ye heard the word of truth, the gospel of your salvation: in whom also after that ye believed, ye were sealed with that holy Spirit of promise,

It is as if Paul knew the Calvinists were coming one day, and so worded this so as to DESTROY the idea of being sealed with the Spirit Himself, which is what regeneration is-John 3:5-8, happens before we believe! Notice twice he says AFTER having already used the word FIRST. They--the Ephesians FIRST trusted in Christ AFTER they heard the gospel, they believed, and AFTER THAT they were sealed with the Spirit, which means he entered their hearts. He is the seal. he regenerates AFTER men believe, not before, which proves men CAN BELIEVE, which disproves Calvinism