Define an n-variable propositional formula as a function $f: \lbrace{0,1 \rbrace}^n \rightarrow \lbrace 0,1 \rbrace$. A function $f$ is a contradiction if it maps all it's inputs to false ($0$ in this notation).
Consider the pair of formula $\phi_1(x) = x, \phi_2(x) = \neg x$. Then $\phi_1$ and $\phi_2$ are themselves not contradictions but $\phi_1 \land \phi_2$ is a contradiction. (We call such a pair and more generally a tuple of formula as almost-non-contradicting, if every subset except the whole set is not a contradiction when taken over a set-wise and).
My question, does there exist a triple $\phi_1, \phi_2, \phi_3$ of $n$ variable formula for some finite $n$ such that $\phi_1 \land \phi_2$, $\phi_1 \land \phi_3$, $\phi_2 \land \phi_3$ are all not contradictions but:
$$ \phi_1 \land \phi_2 \land \phi_3 $$ Is a contradiction.
The following should work for $n=2$:
$$\phi_1(x,y)= x \rightarrow y$$ $$\phi_2(x,y)= y \rightarrow x$$ $$\phi_3(x,y)= \neg (x \leftrightarrow y)$$