Three doors logic problem

The key is the assertion that each door is labelled incorrectly. If capitals stand for the correct labels and lower case letters for the incorrect ones currently present, you will find there are just two possibilities:

$$Ab: Bc :Ca$$

$$Ac :Ba :Cb$$

Each of the doors is labelled differently in the two cases. Whichever door you open, you will therefore know which case applies, and will be able to apply correct labels without opening a second door.

I must confess I cannot follow your notation with the implications and equals. But the key point should be “each door is labeled incorrectly”. Suppose, without loss of generality, that you opened a door which was labeled $A$ but actually contained $B$. Then you know that the door labeled $B$ can't actually contain $A$: if it did, then door $C$ would be labeled correctly, which is against the premise. So you know that the door labeled $B$ must contain $C$ and the door labeled $C$ must contain $A$. This argumentation serves for any door you opened and any possible content, simply by exchanging the names $A$ through $C$.

I don’t follow your solution; words would make it much clearer. Here is one solution.

Open the door labelled Car and Goat.

• If you find only a goat, you know that that the door labelled Car must hide a car and a goat (since it doesn’t hide just a goat and can’t hide just a car); the remaining door, labelled Goat, must hide the car.

• If you find only a car, you conclude similarly that behind the door labelled Goat you’ll find the car/goat pair and hence that the goat is behind the Car door.