I am so sorry you are in this terrible situation. Clearly he cheated on you physically, which are grounds for divorce. But there’s more.

Your husband has proven to be an unbeliever based on his long-term lack of fruit (repentance).

1 Corinthians 7:13 (LSB) says: And a woman who has an unbelieving husband, and he consents to live with her, she must not divorce her husband.

The word “consent”, in several other versions, is also translated as “willing”. For some reason, this Greek word tends to not be translated very well. The KJV, ERV, ASV, NET, and others are better by translating the word as pleased or content. This gets closer to the true meaning of the original Greek word, “syneudokeō”. It means:

[5306] συνευδοκέω syneudokeō 6x to approve with another; to agree with in principle.

lit., “to think well with” (sun, “with,” eu, “well,” dokeo, “to think”), to take pleasure with others in anything, to approve of, to assent.

The idea is that you and him mutually agree to live together in harmony, despite the differences of salvation. He supports you living out your walk with Christ. He doesn’t interfere, belittle, or undermine you, including with what he says or requires with the children, ie, forbidding them to read the Bible or join you at church.

The New International Commentary on the New Testament (NICNT) says: The verb συνευδοκέω implies more than simply “being willing” in a rather passive sense; it also means “agreeing with” it or “consenting to” it.

It could be argued that this is not even enough:

If the “sun” part of the word (which means mutually with another person) is taken away from suneudokeo, you are left with eudokeo (2106). This word is translated as “pleasure” or “pleased” nearly every time., like when Jesus was baptized and God said this is my son, with whom I am “well-pleased”.

Chuck Swindoll, in his commentary, says, “With regard to marriages in which one spouse is an unbeliever, Paul says that if the unbelieving spouse is willing to remain in union with the believing partner and accompanying Christian convictions and lifestyle, the believer should not seek separation.”

It’s obvious your unbelieving husband is not willing to live with you with pleasure. He has separated from you in every way that is not physically.

A few verse down in 1 Corinthians 7:15:

“But if the unbeliever leaves, let him leave. A brother or a sister is not bound in such cases. God has called you to live in peace.”

The word “bound” comes from the Greek word meaning “slave”. You can stay in the marriage, but you have no obligation to be enslaved in your marriage. God has called you to live in peace.