I am trying to prove boolean logic formulas using axioms. I have a lot of trouble deciding what to do, which axiom to use, when to use it, etc. I've asked others on proving formulas but they have all said that the only way to solve them is by brute force or guessing.
Is this true? Is there a pattern to proving boolean formulas or things to look for?
As an example, I am given the following formula. (Please, don't mind the formatting of implication, equality, etc. I couldn't find the correct symbols.)
$X \rightarrow (Y \rightarrow Z) = (X \rightarrow Y) \rightarrow (X \rightarrow Z)$ (implication theorem)
$X \rightarrow (\lnot Y \lor Z)$ (distributivity of implication)
$(X \rightarrow \lnot Y) \lor (X \rightarrow Z)$ (implication theorem)
$(~X \lor ~Y) \lor (X \rightarrow Z)$ (De Morgan's Law)
$(X \land Y) \rightarrow (X \rightarrow Z)...~(X \land Y) \lor (X \rightarrow Z)$ (implication theorem)
I have gotten this far and don't know what to do next. Am I even on the right track?