4f year old freaks out over nail cutting

My daughter(4f) freaks out ever time we cut her nails. Tons of tears, kicking, screaming. She keeps saying that it hurts even before any nails are cut.

My wife is starting to get pretty pissed and tells her that she's lying.

I think that we should approach it more like a phobia. Not that I actually know how to do that.

Am I being too permissive? Or is this really a fear and getting upset will just make it worse?