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I am extending off of a previous question NAND, XOR and AND, and I am attempting to prove that using XOR, AND and constant True can be written to NAND. This is because I was already able to prove NAND with A∧B, A∨B and ¬A.

Below is the proposition that is equivalent to NAND and I am sadly stuck on where to go now. I was thinking of expanding the last XOR but I feel that I can reduce the left side more before that?

[{A V B} ⊕ {A ⊕ T}] ⊕ A

Here is the work I have done so far,

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Your expression uses an $\lor$, so that is definitely not good, since you need to use only $\land$, $\oplus$, and $\top$.

Here is something you can do: since $A$ NAND $B \Leftrightarrow \neg (A \land B)$, and since $P \oplus \top \Leftrightarrow \neg P$, you have that $A$ NAND $B \Leftrightarrow (A\land B) \oplus \top$.

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    That was suppose to be a AND, so it was a typo, thanks for pointing that out – 2017-01-22