Say that $a_1, \cdots, a_{n-1}$ is an independent generating set for $S_n$. Let $b$ be any element in $S_n$. Is it true that $b$ can replace one of the generators, i.e. that there exists an index $i$, such that we have that $a_1,\cdots, \hat{a_i},\cdots, a_{n-1}, b$ generate $S_n$?

If $a_1, \cdots, a_{n-1}$ is the standard (n-1)-tuple that generates $S_n$, $(12),(13),...,(1n)$, then it's true and it can easily be shown. Does it hold in general?