I try to solve the following exercise from a textbook and need some help:
A set $\Phi$ of S-sentences is called independent if no $\phi \in \Phi$ is a consequence of $\Phi - \{\phi\}$.
a) Every finite set $\Phi$ of S-sentences has an independent subset $\Phi_0$ such that $Mod_S\Phi = Mod_S\Phi_0$.
b) If S is at most countable then every $\Delta$-elementary class of S-structures has an independent system of axioms. (Hint: Start by defining an axiom system $\phi_0, \phi_1, \dots$ such that $\models \phi_{i+1}\rightarrow\phi_i$ for all $i\in \mathbb{N}$.)
I solved a) but don't have a good idea for b). Any ideas?