In "A Concise Introduction to Mathematical Logic," the author, Rautenberg, provides a proof that propositional logic is complete. This proof first shows that any consequence relation satisfying the deduction rules is maximally consistent (a consequence relation is consistent if there is a set of of formulas from which not every formula is provable). I'm able to follow the proof, but am not able to pick up any intuition at all for what's going on. Can somebody help explain "in broad terms" what's going on intuitively here? The proof (bottom image) is pasted below as the solution to an exercise (top image).
My current understanding is that for any consistent consequence relation there is a consistent set of formulas $X$ which can be extended to a still consistent set $Y$. Then we can do this substitution where everything in $Y$ becomes trivially true (provable) and everything outside of $Y$ trivially false. Then what?
