If you would like legal advice about your specific situation, then you will need to retain an attorney. But honestly, it's probably cheaper to fix the AC yourself rather than hire a lawyer to sue the seller for you.
In any event, here is just some basic principals on contracts. Most contracts do not need to be in writing. If there is a dispute about what the terms of the contract are, a judge/jury would need to hear the facts and make a determination. If the trial reveals that the condition about the AC was part of the contract, then it is just as good as if written down. The seller makes a valid offer if the offer includes sufficient required terms (like price, quantity, identification of goods, etc.). Depending on the type of contract and the relationship of the parties, the offer may need many specifics, of not many at all. If the buyer accepts, then they have a deal. If the buyer adds any material condition or changes to the offer, then it is considered that the offer is rejected and the BUYER is now making a new, counter-offer to the seller. If the seller accepts, then they now have a deal that includes everything in the counter-offer. The seller cannot accept the counter-offer and renege on the conditions, or else he will be in breach of contract.
Hope this helps. Good luck!
This posting is just general legal INFORMATION and not legal ADVICE. Only your attorney can provide legal advice. If you would like actual advice about your particular case, please contact me for a consultation.
Yes, there is such a concept "acceptance" by silence or conduct. Your questionsreminds of a law school essay fact pattern. Argue that the fixing the AC was a condition precedent to the contract. In other words, there is a contract only if he fixes the condition first. You can file in small claims court.
This answer is for informational purposes only and should not be construed as legal advice.
While is might be cheaper to fix the A/C, the technical answer is you made a counter office and by taking the payment it was conditioned up the defendant to fix the A/C. This fact situation could be a test question in a law school Contract class. Good luck