# An equivalent of $\neg ((a \wedge b) \Leftrightarrow c)$ based on a specific requirement

I am solving a problem that is asking to provide an equivalent for $$\neg ((a \wedge b) \Leftrightarrow c)$$ and the equivalent shouldn't contain $$\wedge$$ (and), $$\Rightarrow$$ (implication) or $$\Leftrightarrow$$ (iff). What I have worked out is:

$$\neg(\neg((\neg a \vee \neg b) \vee \neg c) \vee \neg(\neg(\neg a \vee \neg b) \vee c))$$

I wonder based on the requirement stated above, is there a shorter equivalent for my answer?

Update

$$\oplus$$ is not allowed in the requirement either.

Outgoing from requirements can you use $$\oplus$$? if yes, then, as we know, that $$\neg(a\Leftrightarrow b) = a \oplus b$$, then we can obtain $$\neg ((a \wedge b) \Leftrightarrow c) = \neg (\neg a \lor\neg b) \oplus c$$
Accordingly to your update I wrote my one: not shorter, slightly different, but logically equivalent yours one $$\neg \big[(a \wedge b) \Leftrightarrow c\big] = \\ =\neg \big[\big((a \wedge b) \Rightarrow c \big) \land \big(c \Rightarrow (a \wedge b)\big) \big]=\\ =\neg \big[ \big(\neg (a \wedge b) \lor c \big) \land \big( \neg c \lor (a \wedge b) \big) \big]=\\ =\neg \big[ \big(\neg a \lor\neg b \lor c \big) \land \big(\neg c \lor \neg(\neg a \lor\neg b) \big) \big]=\\ = \neg\big(\neg a \lor\neg b \lor c \big) \lor \neg\big( \neg c \lor \neg(\neg a \lor\neg b) \big)$$