Consider we have the following statement to prove: $P \implies Q \wedge R$. For a proof by contradiction, we assume $P \wedge (\neg Q \lor \neg R)$.
How would one go about this? Typically to prove a statement of the form $(A \lor B) \implies C$, we show $A \implies C$ and then $B \implies C$. Do we break down such a proof by contradiction into such cases?
Does it suffice to reach separate contradictions by considering the cases $P \wedge \neg Q$ and $P \wedge \neg R$ separately?
Must one also consider the case when $(P \wedge \neg Q)$ and $(P \wedge \neg R)$ are both assumed to hold "simultaneously"?
Many thanks in advance for your help.