How do we make sure in Paxos that we don't propose a different value if a majority has formed?
$P2^c\implies P2^b$ since $P2^c\iff (a)\lor (b)$ and the logical statement $(a)\lor(b) \implies P2^b$ is equivalent to $(a)\implies P2^b\land (b)\implies P2^b$.

Let's consider $(a)$. If no acceptor in $S$ has accepted anything below $n$, then $P2^b$ is trivially satisfied (the implication predicate is false).

Otherwise, assume $(b)$. Then there exists a highest-numbered accepted value $(v, n)$ for some acceptor within $S$, which the proposer will decide to send as an accept request. Thus $v$ must have been chosen already, since an acceptor had it.