The Abrahamic Covenant and Infant Baptism

In my continuing reading of "The Gospel Covenant", by New England Puritan Peter Bulkeley, I am currently at a portion in which he defends the identity of the Abrahamic Covenant with the New Covenant in substance, while distinguished in administration. He says (archaic spelling and grammar in original):

"This poynt, concerning the identity or sameness of the two covenants, doth lay a good foundation for communicating Baptisme to the infant children of believers. For if both these things be true, first that the old and new Covenant be in substance the same; and secondly, that children are within the new as they were within the old, then there can be no sufficient reason to deprive children of the seale of the Covenant now more than former times under the old; and, that argument drawne from Circumcision to Baptisme, will stand against all the batteries which are made against it, never to be beaten downe whiles heaven and earth doe endure. The Covenants are the same, and the signes of the Covenants (Circumcision and Baptisme) are in signification the same also; and the children of the faithfull have the same relation and right to the Covenant now as they had before; What reason then that children being before circumcised, in token of their being in covenant, should be forbidden to be baptized , that it might be to them a signe of the Covenant betwixt God and them? It is even a wonder of wonders, that in such cleare light so great mists should be raised up to darken the truth. Let humble mindes search the truth in love, and the Lord will reveale the same unto them."

The verses Bulkeley refers to are Genesis 17: 7, "I will establish my covenant between me and you and your offspring after you throughout their generations for an everlasting covenant, to be God to you and your offspring after you," leading to verse 10, the institution of circumcision as the sign of that covenant, administered to each male infant on his eighth day. Then he points to Paul's use of verse 7 in Galatians 3:16 (see also Romans 4), which he then relates to baptism in verse 27. Then in verse 29, Paul concludes, "And if you are Christ's, then you are Abraham's offspring, heirs according to the promise."

So, Bulkeley's logic proceeds like this:

1) God's covenant promise to Abraham explicitly included his children, who therefore received circumcision as the sign of the Covenant.

2) Christians are the children of Abaraham by faith, and therefore receive His covenantal promises.

3) One of those promises is to bless our offspring.

Therefore, our offspring have a right to receive the sign of the Covenant, i.e., baptism in the place of circumcision.

In the debate over the proper subjects of baptism, credobaptists demand an explicit commandment to baptize the children of believers. Rather, given Bulkeley's argument here, it is actually incumbent on the credobaptist to show an explicit commandment that children are not to receive baptism.