Generally, your opening was correct, but you did not use chrono boost effectively in the early game at all. You also want to to avoid getting a Stargate before any sentries. Also, don't open two zealot when doing this build; instead zealot/stalker/sentry is the way to go. When you scouted your enemy, you noticed he had nearly 3 chrono boosts, 1 gas, and a zealot building. With this information, you should be weary of a gateway rush, as it was aiming to be that way. Additionally, your probe can stick around and continue to scout until that stalker is about halfway built, then you may run away safely. Go hold the middle tower with a zealot or your probe so that you know this isnt coming. This will allow you to opt out of building the second sentry, simply because you know you are safe; allowing for more gas for Robo and Stargate.
When opening Phoenix, it's wise to chrono boost out two as fast as possible; some players scout on the first, some wait for the second, because you can snipe two gas probes and take the information and be on your way. I think either or is okay, because the first phoenix will still gather all the valuable information you need. Also, when executing this the purpose is containment and information. You already know this, but you built just too many phoenix. 4-5 will do the trick just nicely. You also need to be a little more active with them; Phoenix are the fastest air unit in the game, so make sure to use them that way. Its incredibly easy to drive by snipe sentry's and probes.
Also, when transferring probes make sure that both nexus are rallied to the expansion if your main is already saturated; you want to saturate your exp as quickly as possible. You also expanded much more effectively against him, so you need to get your gas on the expansion faster and advance your tech. You actually almost lost to his push because you simply had too many Phoenix, and Immortals and not enough AoE and Zealots. You don't need many immortals to deal with too many Stalkers as the Colossus are already going to do a fantastic job. By overbuilding phoenix and immortals you left a window open to be killed. Most players will go directly for colossus tech, because it simply helps them defend better, and will be great against stalker, zealot, immortal, archon. In the main fight you will be lifting immortals and stalkers, and your own 3 Immortal/3 Colo army with gateway units will shred him apart. Also, keep a observer with your army, as he was watching you a good amount of the time. After the second base is up and running throw down a second Robo and maintain an AoE advantage over his Anchons and make sure to have enough zealots to tank enough damage so your colosus remain untouched. Remember you do not need anymore phoenix beyond the 4-5 originally.
You need to learn to extend your leads. Next time your in this same position throw down a double robo faster after expansion, and you could of easily maintained the upgrade advantage the whole game; your forges were not active enough.
Always, always, always be spread apart in defensive posture. (spread out) You want to aim for a 8 Colossus army, then begin to add a meat shield of immortals, then its a small group of stalkers, and zealot/archon. If the game goes super late you add a mother ship into the mix. If minerals begin to float close to a 200 army you just expand and add gateways. Against compositions such as his, dont be shy to build half-gateway walls to use a defensive choke point against units such as zealot/immortal/archon push. Its a staple defensive method, and once you get enough colo, the game is yours unless he does something drastic and you make a colossal mistake.
1. Zealot/Stalker/Sentry opening begin second sentry and cancel it if nothing is coming your way. Remember to hold the Xel'Naga tower. When cross positions' I can't stress how important it is. It may allow you to skip a step or two. Keep the Stalker/Sentry close to home to prevent scouting and defend.
2. Chrono boost for the love of Auir! Probes, units anything, But especially probes. Full chrono boost is pretty much always bad.
3. Do not over build Phoenix and only expand when you know your safe. You moved out before you knew what was coming.
4. Do not over build immortals. Colo/Zealot/Phoenix/Immortal is going to hold a Zealot/Immortal/Archon/Blink Stalker army any day. Control the immortals with phoenix and position the rest of your units properly. Even if he wipes our your meat shields, the colo advantage should pull through with the back up of your gateways.
5. You have to try and pay out more with the Phoenix focus on this, there was a lot of nice openings and gaps you could of exploited. You will get better with multitasking.
6. Faster second Robo after expansion, and make sure to double assimilator your expansion ASAP. Gas is too important in PvP.
7. You expanded faster, and teched harder. You should of been ahead in upgrades, and higher tech units, instead you were only 2-2.
You build probes through the game, which is good. You need to make sure your observation improves so you know how to pounce on and extend leads. By cutting down 1-2 phoenix and 1-2 immortal you already have another Robo bay down producing double colo. You should try to get this as reasonably fast as possible toward a saturated income and 4 gas.