Walk me through an IPO valuation for a company that's about to go public?

The guide says that after you find comparables and use a relevant multiple to estimate the company's Enterprise Value, you need to subtract the IPO proceeds when backing into Equity Value. The guide is very vague in this context; surely, you are not supposed to outright subtract this "new" cash from the Enterprise Value, thereby decreasing Equity Value, right? I thought the guide maybe was maybe referring to subtracting the new cash from the existing debt balance, thereby decreasing net debt and increasing equity value when isolating that variable in the EV equation (Equity Value = Enterprise Value - net deb).

However, the guide also says that if you use an equity value-based multiple (i.e. P/E), then you would arrive at Equity Value instead of Enterprise Value. The guide states, "then you would get to Equity Value instead, and subtract the IPO proceeds from there." I am having a really tough time understanding why you would need to subtract the IPO proceeds. Wouldn't the Equity Value you calculate already represent value attributable to all equity investors? Why do you need to subtract the "new" cash?

Thanks in advance for the help.