If you only need cold water, why do you want to connect the hot and cold supply lines together ?
You only need to cut in a repair branch fitting on the cold supply and run a pvc line over to the washer location. Make sure you cut the line square leaving a gap equal to the branch line id, clean off the burr, clean and prime the line and fitting, then apply the pvc cement to both the fitting and the pipe. You need to give it a quarter turn twist and line up in the correct orientation. If you can’t get a repair fitting, you will need to be able to move the pipe to one side or both the full length of the fitting while gluing. If that is not possible, cut a larger section out and use the branch fitting, a short piece of pvc pipe and a repair coupling. Glue the repair coupling in place last.
PVC glue joints do not like to be be under stress. The “glue” will creep and the glue joint can fail.
i assume you will be using a hose to connect to the washer cold supply valve ?
What about the plans for the drain line for the washer ?
Most codes require an air break to prevent siphoning from the drain line back into the washer.