The Tridentine attribution is erroneous, although it continues to pop up over and over and over. Marriage for everyone is regulated by the Natural Law, for the Jews by the Mosaic law, for Catholics by the moral component of both, the divine law, and the requirements imposed by the Church on Catholics by the Power of the Keys.

For validity the basic requirements are a couple, male and female, with no diriment impediments (mental capacity, not under duress, no existing marriage, etc).

Christ abolished the requirements of the the Mosaic law and instituted the divine law of monogamic and indissoluble marriage. He also raised marriage to the dignity of a sacrament (Genesis 2:24; Matthew 19:3 sqq.; Luke 16:15 sqq.; Mark 10:11 sqq.; 1 Corinthians 7:2 sqq.)

The requirement for a priest as witness is a disciplinary one, can be waived, and is waived where priests are rare. The priest as the witness of the Church has appeared in different places over the centuries.