What exactly do you want proof of?
That he's a twat and a shit dad? Here it is:
Our relationship has struggled since DD was arrived, mainly because DH can't seem to cope with not being centre of attention anymore and because my priorities have changed from going out all the time with him to wanting to spend time at home with DD and trying to catch up on sleep.
That he's a slimy little cheat with no repsect for you? Every line of your post tells me that.
I think you need to take power here, properly take it.
This is a second-rate man who is dragging you down. He doesn't support you, he does not make you feel loved, he has cheated before and clearly continues to interact disloyally with other women - it's only a matter of time before he 'technically' cheats again if he isn't already. Above all, he's a poor father who puts himself and his wants above the needs of his DD.
You KNOW you would be better off without him.
So forget about 'proof'. What you mean by that is, how can I build a case which will allow me to walk away? - ie, I need a cast iron reason to leave this man, I'm not allowed to just decide that he's shown himself to not be good enough. I need to convince HIM I've got a good enough reason otherwise I know DH will turn it around to me and all he will do is complain that I went through his phone.
In other words, DH is 'in charge' of this relationship. I need to get him bang to rights or my decision won't be good enough, and I'll have to stay.
Fuck that!!
Take that power. You can leave this man simply because you want to. Because he's not good enough and you don't love him, and that's his fault.
'DH, it's over. I don't love you and I don't want to be with you any more.'
Don't even tell him about going through his phone. What business is that of his? He is dismissed. There is no argument to be had. He's been judged on many elements of his behaviour and found completely wanting, so he is gone.
Yes he will rant and rave, then plead, then rant and rave some more, demand reasons (you, blandly - 'I'm so sorry, I guess my love for you has just been eroded over the years, but there's no point in dragging all that up now.') - Then he'll threaten to take your child away - 'I'm going for 50/50 custody!!!' (You: 'Oh that will be wonderful, I was afraid you wouldn't want to give your full support, that's great, we will both need time child-free to build new lives and relationships so 50/50 is exactly what I want' - Him: 'Fuck off you bitch I'm not babysitting so you can go fuck other men, you'll be having custody!' - You - sweeeet :) ) And rinse and repeat that a few times and then he will fuck off and you will be free.