That's not really what happened. Nintendo was working with Sony to create a CD add-on for the SNES. As a part of that deal Sony was making a version of the SNES that included the CD drive add-on called the PlayStation similar to how Sharp made the Twin Famicom that incorporated the Famicom Disk System. Nintendo realized the deal with Sony would give Sony too much control over CD games so they scrapped it and went with Philips (which is how they got to make those bad Nintendo CDi games). In the end a SNES CD add-on was never released.
Sony was pretty pissed off when Nintendo announced they were going to be partnering with Philips when they thought Nintendo was going to be announcing they were working with Sony. So they ended up entering the gaming market to compete with Nintendo. However the PlayStation they came up with was an entirely new design focused on 3D graphics.
Just to add, the CD add-on to the SNES would have added CD-based audio and presumably larger/cheaper games, but would have added no additional computing power. None of the playstation hardware was based on this product.
Sony was pretty pissed off when Nintendo announced they were going to be partnering with Philips when they thought Nintendo was going to be announcing they were working with Sony. So they ended up entering the gaming market to compete with Nintendo. However the PlayStation they came up with was an entirely new design focused on 3D graphics.