No.
The north would not have invaded if there had not been slavery
You are skipping a lot of steps to go from your assertion to your conclusion, and your conclusion is deliberately misleading.
The North didn't invade to stamp out slavery. The North invaded to stop the South from becoming independent. Why did they North refuse to let the South go? Was it out of principle or something?
Lincoln was going to let the original 7 states leave if Virginia would pledge to remain in the Union. So okay, no Principle reason to "preserve the Union."
Well what was it then? Why would Lincoln want to keep a bunch of slave states in his Union? Why did he insist he had no interest in freeing the slaves for nearly two years into the war?
What's left?
Money. The South was a cash cow fueling New York and Washington DC pockets. So long as it remained in the Union, they could keep the slavery produced money flowing into New York and Washington DC.
If it got out, that money not only stopped, that money would then fuel competing industries in the South with existing Northern manufacturers.
Why did the south want to secede?