I accidently got a 700x20 front tire and a 700 x 23 rear tire. Will I have any problems with this?
No, you won’t have any problems.
In fact, lots of people do that on purpose 
I accidently got a 700x20 front tire and a 700 x 23 rear tire. Will I have any problems with this?
No, you won’t have any problems.
In fact, lots of people do that on purpose