Your wife gets angry because of the overwhelming state of uncertainty and doubt. Parenting is a tough job sometimes because, unless we learn to parent effectively, we usually parent the way we were parented. Imagine that your child isn't going through a rebellious phase, but actually, communicating. Imagine your child is using his behavior to communicate something his words are not.
Perhaps imagine that your child is actually saying "you can't make me!!". What your child is telling your wife is "give me choices, involve me usefully". Misbehavior is communication. Your child is seeking significance and belonging which are his emotional needs. His "no" is not something to be broken, rather to be appreciated as a message. Ask him what he wants by giving him choices from two things. For example, instead of PUT YOUR SHOES ON. "what do we put on our feet before going outside?" or "what do you want to wear the sandals or sneakers" (can you see that wearing the shoes is a message within the choice?)
Stop all threats, time out, and punitive consequences (punishment). Things will continue to worsen by using those. Spend more one to one time with this child. That means that you spend at least 10 minutes twice per day of totally present, uninterrupted, focused, involved, time, doing what he wants to be doing, (playing, reading, building, walking, etc.) that doesn't involve any phones ,tv, or electronic devices. Establish that ritual.
When your child gets to that rebellious point, he is rebelling against something or someone, right? Ok, so if you want to take away something, take away the reason for rebellion. How? When you see him starting to get "there"...In a very calm voice... Tell him that you need a hug. Wait without saying another word. A few moments later, tell him again. If he still didn't want to, tell him a third time. If again he refuses, tell him that you need a hug, so when he feels like giving one, to come find you, and that YOU will be going to YOUR quiet space. DISCONNECT quickly, while you still have your wits about you, and don't become reactive, because then boom!
Next, get a life outside parenting. Exercise, hobbies, volunteer, connect with you. Think, what do you want? What do you have? What do you need? How many ways can you get it? What are you doing? Answer those questions to help you determine something outside of motherhood. It will help you become more leveled.
The idea is to become your child's Leader, not Overseer or Commander. Obviously, you can tell already that overpowering isn't effective. Think on this. His rebellion is a result of him feeling discouraged. He is feeling insignificant because he is not permitted to discover his innate desire for autonomy. He is feeling discouraged because he has no positive connection, just correction, and threats, so his attention is negative.
By changing the dynamic with one to one time, disconnecting, hugs, choices, and connection, as well as, getting in touch with yourself... You will get very different behavior. Training is another thing to do, but I would establish the encouraging and loving relationship again first. Best to you and your family.