Pretty straight forward question, every time I hear about people wanting minimum wage to go up I keep thinking "Wouldn't the price of everything just increase?"
I would suggest looking at the history of minimum wage increases. As far as I'm aware, there isn't a correlation between minimum wage hikes and a lagging inflationary period. Empiricism trumps theorizing about what "should" happen. Or, as stated by Hume, "You can't get an ought from an is."