Federal Headship
In the latter half of Romans 5, Paul teaches that the entire human race is summarized in the two Adams. The first Adam was the federal head of the race under the covenant of works; the second Adam, the Lord Jesus Christ, is the federal head of all believers under the covenant of grace. Thus, as the sin of Adam was legally and effectively our sin, so the obedience of Christ is legally and effectively the righteousness of all believers. The federal relationship in which Adam stood to the race was the ground of the imputation of his guilt to them and the judicial cause of their condemnation. And the law that condemned them could not justify them unless an adequate reparation should be made for the wrong done, a reparation they were incapable of making because of the corruption they inherited from Adam as their natural and federal head. To provide their salvation, the needed reparation had to be made by another who was not of federal connection with Adam and therefore was free from the imputation of his guilt. Federal theology represents these requirements as being met in Christ, the second Adam, in whom a new race begins. God had entered into covenant with him, promising him the salvation of all believers as the reward of his obedience. But the obedience required of him as the federal head of his people was more than the mere equivalent of that required of Adam. His representative obedience must include a penal death. And thus his resurrection victory is also the victory of the new humanity that has its source in him. (Evangelical Dictionary of Theology, s.v. “Federal Theology.”)
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By Scripture
Old Testament