“Sec 121. Bigamy consists in the having of two wives or two husbands at one and the same time, knowing that the former husband or wife is still alive. If any person or persons within this State, being married, or who shall hereafter marry, do at any time marry any person or persons, the former husband or wife being alive, the person so offending shall, on conviction thereof, be punished by a fine, not exceeding one thousand dollars, and imprisoned in the penitentiary, not exceeding two years. It shall not be necessary to prove either of the said marriages by the register or certificate thereof, or other record evidence; but the same may be proved by such evidence as is admissible to prove a marriage in other cases, and when such second marriage shall have taken place without this state, cohabitation in this state after such second marriage shall be deemed the commission of the crime of bigamy, and the trial in such case may take place in the county where such cohabitation shall have occurred.”

Revised Laws of Illinois, 1833, p.198-99

This wouldn’t apply to Fanny Alger or the Johnson girl who were pre-Nauvoo, but this was the law in effect for most of Smith’s marriages.

Source for the law was based on English law:

at common law, the second marriage was always void (2 Kent, Com. 79), and from the earliest history of England polygamy has been treated as an offence against society. After the establishment of the ecclesiastical [98 U.S. 145, 165] courts, and until the time of James I., it was punished through the instrumentality of those tribunals, not merely because ecclesiastical rights had been violated, but because upon the separation of the ecclesiastical courts from the civil the ecclesiastical were supposed to be the most appropriate for the trial of matrimonial causes and offences against the rights of marriage, just as they were for testamentary causes and the settlement of the estates of deceased persons. By the statute of 1 James I. (c. 11), the offence, if committed in England or Wales, was made punishable in the civil courts, and the penalty was death. As this statute was limited in its operation to England and Wales, it was at a very early period re-enacted, generally with some modifications, in all the colonies. – U.S. v. Reynolds, 98 U.S. 145, 164-65 (1878) (emphasis added).

The 1862 Morrill Anti-Bigamy act, the 1879 SCOTUS Reynolds decision, and the 1882 Edmunds Act all reaffirmed the illegality of Mormon “plural marriage.”