expanding requirements for equivalent conditions We have all seen statements about equivalent conditions, such as If any one of the following three conditions hold, then all three conditions hold.
Are there any examples of three conditions which all hold, provided at least two of them hold? So to be slightly more concrete, given conditions (a), (b), and (c), we know that if (a) & (b) are true, then (c) is true, but if either (a) or (b) is false, then (c) need not be true. This formulation would hold up to any permutation of (a), (b), and (c).
I've given this some thought, but I really have no clue how to approach this question with rigor, although if there were a way, I feel it would be quite simple. I thought perhaps someone would know a way to demonstrate the existence or impossibility of this situation or have an explicit example.
 A: Let $ABC$ be a triangle.


*

*$ABC$ is an isosceles triangle.

*There exists a right angle in $ABC$.

*There are two angles of $45^\circ$ degrees.


Every two imply the third, but one alone is not enough to imply any of the others.
A: Let $V$ be an $n$-dimensional vector space. Let $S$ be a subset of $V$. 
(a) $S$ is a linearly independent set. 
(b) The span of $S$ is $V$. 
(c) The number of elements of $S$ is $n$. 
Any two imply the third. No one implies either of the other two. 
A: I love this simple one. Let $A$ be a set, $\sim$ be an equivalence relation on $A$, let $x_0\in A\land x_1\in A\land x_2\in A$.


*

*$x_0\sim x_1$.

*$x_1\sim x_2$.

*$x_2\sim x_0$.

A: Let $X$ be a topological space.


*

*$X$ is finite.

*$X$ is discrete.

*$X$ is compact Hausdorff.


No one implies another, but any two imply the third.
A: A generalization inspired by mjqxxxx's comment and Shamir's Secret Sharing:
Let $f(x), g(x)\in K[x]$ be polynomials of degree $d$ and suppose $x_1, \dots x_m$ are distinct elements of the field $K$ for some $m> d$. Then any $d+1$ of the following statements together will imply any (all) of the others.
$$1.  f(x_1) = g(x_1)$$
$$1.  f(x_2) = g(x_2)$$
$$\dots$$
$$1.  f(x_m) = g(x_m)$$
