Prove that $(p \rightarrow q) \land (\lnot p\rightarrow q)$ is logical equivalent to $q$ by using a chain of logical equivalences.
The question states explicitly not to use a truth table.
I tried the following: $(p \rightarrow q) \land (\lnot p\rightarrow q)\equiv (\lnot p \lor q) \land (p \lor q) $. Distributing the operators, we obtain: $(p \rightarrow q) \land (\lnot p\rightarrow q)\equiv q \lor (\lnot p \land q) \lor (p \land q) $.
However, I don’t know how to continue. Please help!