Some players may be surprised that the Robotics Facility is not the best option for this opener. While Immortals buy ff14 gil are a great unit with heavy Sentry support, they are not so great on their own. Good Zergs are not going to simply let the Immortals waltz across the map and join your forces at your proxy Pylon.
A good Zerg will place a pack of Zerglings between your main base and the proxy Pylon and intercept units running across the map. Zerglings can easily trap and surround Immortals, taking them out before they ever make it. Even when the Immortal does make it, it can be focused down by Zerglings if it does not have a bunch of Sentries to support it (you will not have the gas for mass Sentry). In the lower leagues where Zergs may not do this, so a Robotics Facility can work.
If you want to use Immortals with this timing, you need a Warp Prism, as the Warp Prism can pick up the Immortal and safely carry it to your main fighting force at the proxy Pylon. The Warp Prisms can also constantly pick up and drop the Immortals to prevent the Zerg from focus-firing them. This requires excellent micro.
Stargate units do not have these problems. They fly, so they do not have to be worry about being intercepted by Zerg tier 1 units as they run across the map. Phoenix in particular are very fast so they reinforce quickly, making them popular for this build.
With the early gas build, you likely can pump out 2 Phoenix before your Gateways finish up and you are ready to attack. With the early Nexus build, you will likely move out with ground units and make your initial attack, rallying your air units to join you later in battle.
Once you get to this point, the rest is straight forward. Attack your opponent, using primarily Zealots from the Gateways and Phoenix from your Stargates. Try to keep all of your Phoenix alive so the count can build up to 8 or more. Once you have 8 Phoenix, the game is over. You can keep the Zerg supply blocked constantly by ignoring Spore Crawlers and popping Overlords after they spawn. Phoenix also pick up Queens and Roaches as these units will kite and annoy your Zealots. The Zealots can provide DPS against Zerglings, Drones, Spore Crawlers, and buildings. Once your Phoenix count gets up to a very high level, you may switch over to Void Rays.
You should continue to produce Probes at home so you have something to fall back to should your attack not win the game outright. You may also drop a Robotics Facility right before your main attack (only after getting the third and fourth Assimilators up and running) just to unlock the tech. This will make it easier to move up to Colossus tech should you need to fall back.
Falling back is okay as long as you destroy the Zerg third while building enough Probes to saturate both your bases. A Protoss player on two fully saturated bases and tier 2 tech is well ahead in a game against a Zerg player on the same number of bases with similar or less technology.