The intuitionistic propositional formula $\mathbf{bb}_n$ (in the $n+1$ variables $p_0,\ldots,p_n$) is:$$\bigwedge_{i=0}^n \Big ( \big (p_i \Rightarrow \bigvee_{j\neq i} p_j\big) \,\Rightarrow \, \bigvee_{j\neq i} p_j \Big) \; \Rightarrow \; \bigvee_{i=1}^n p_i$$
In Chagrov & Zakharyaschev, Modal Logic (1997), proposition 2.41, it is proved that $\mathbf{bb}_n$ holds in a finite Kripke frame iff the latter has branching $\leq n$ (in the sense that every node has at most $n$ immediate successors).
QUESTIONS: What is the first appearance of this formula in the literature, and who first proved this result and where? Was this formula ever considered in any other context than this particular result?
I am also interested in any potential sighting of the weaker variant (maybe call it $\mathbf{bb'}_n$)$$\bigwedge_{i=0}^n \Big ( \big (p_i \Rightarrow \bigvee_{j\neq i} p_j\big) \,\Rightarrow \, \bigvee_{j\neq i} p_j \Big) \, \land \, \bigwedge_{i=0}^n \Big ( \neg p_i \Leftrightarrow \bigwedge_{j\neq i} p_j \Big ) \; \Rightarrow \; \bigvee_{i=1}^n p_i$$in the wild (I don't know if the finite frames satisfying it can be characterized in a simple way).
The main reason I ask is that $\mathbf{bb'}_2(\neg q_0,\neg q_1,\neg q_2)$ (i.e., the formula obtained by replacing $p_i$ by $\neg q_i$ in $\mathbf{bb'}_2$) was shown by Valery Plisko in 1973 to be realizable (“Ореализуемыхпредикатныхформулах”, Докл. Акад. Наук212 553–556, where it appears as “$L$” on page 555) and I'd like to know whether the $\mathbf{bb}_2$ formula was already floating around at that time.