Need to use deduction to show $(p \lor q) \land (\lnot q \lor r) \implies (p \lor r)$ is a tautology

Question

$$(p \lor q) \land (\lnot q \lor r) \implies (p \lor r)$$

My work so far:

I feel like I'm doing something very wrong and making the problem get extremely tangled

Answer 1

We should make use of the Implication laws directly. It simplifies the process.

Additionally, we should use this rule. Conjunction implies disjunction. $$\begin{align} (a \wedge b) \implies (a \vee b) & \space \space \space\text{[E6]} \end{align} $$

The deductive proof scheme is as follows. $$\begin{align} & (p \vee q) \wedge (\neg q \vee r) \\ 1. \space & (p \vee q) \\ 2. \space & (\neg q \vee r) \\ 3. \space & (p \vee q) \vee (\neg q \vee r) \iff (p \vee r) & \text{[E6, Associativity]}\\ \end{align} $$