Raising the minimum wages results in higher wages, even when compared against inflation

The overall purchasing power of those in the working class (lower & middle class generally) increases faster and higher than the rate of inflation. So, while the nominal price of goods & services would increase, the real price decreases for most people because their wages increase at a faster rate. The people who would see a real increase in prices are those with high enough wages (the wealthy) whose wages would not nominally increase due to an increase in minimum wage.