John 6:63 It is the Spirit who gives life; the flesh is no help at all
John 6:63: This is why I told you that no one can come to me unless it is granted him by the Father.
1 Cor 2:12, 14 no one comprehends the thoughts of God except the Spirit of God. Now we have received not the spirit of the world, but the Spirit who is from God, THAT WE MIGHT UNDERSTAND the things freely given us by God...The natural person does not accept the things of the Spirit of God, for they are folly to him, and he is not able to understand them because they are spiritually discerned.
Yes, the Spirit gives life. When we believe. He does not give life so we can then believe.
One problem with systematic theology is that it takes this statement:
“I am the bread of life; whoever comes to me shall not hunger, and whoever believes in me shall never thirst. But I said to you that you have seen me and yet do not believe. All that the Father gives me will come to me, and whoever comes to me I will never cast out. For I have come down from heaven, not to do my own will but the will of him who sent me. And this is the will of him who sent me, that I should lose nothing of all that he has given me, but raise it up on the last day. For this is the will of my Father, that everyone who looks on the Son and believes in him should have eternal life, and I will raise him up on the last day.”
and attempts to dissect it. Neither God nor the Gospel is a frog to dissect. God draws (which does not mean compel) and whoever comes will not be rejected.
But we do know some things explicitly. Life follows faith. “If anyone thirsts, let him come to me and drink. Whoever believes in me, as the Scripture has said, ‘Out of his heart will flow rivers of living water.’”
How and to what degree God reveals himself to each individual man is something we do not know. God doesn’t tell us. But we know that God’s wish is for all men to be saved, we know it will not happen, and we know that to those who hear the Gospel, salvation comes to those who repent and believe.