[Math] Propositional Logic Help: $(\neg p \wedge (p \vee q)) \rightarrow q $ is a tautology

Question

I need to prove that $(\neg p \wedge (p \vee q)) \rightarrow q $ is a tautology using Laws of Logic (not truth tables).

This is what I tried:

$\equiv (( \neg p \wedge p) \vee (\neg p \wedge q)) \rightarrow q \\
\equiv (F \vee (\neg p \wedge q)) \rightarrow q \\
\equiv (\neg p \wedge q) \rightarrow q \\
\equiv (F) \rightarrow q \\
\equiv T $

Is this logically correct?
The laws I used in order were: Distributive, then negation, and identity. My only issue is with the last step where I know the truth values of $(\neg p \wedge q)$ are all $F$ but I dont know what law it uses.

Please Help!

Answer 1

Hint: We have that $$ (\neg p \land q)\to q \equiv (p\lor \neg q)\lor q \equiv p \lor (\neg q\lor q). $$ You should be able to take it from there.