This is a follow-up inquiry to If the antecedents are mutually exclusive, then is the consequent true?.
Suppose I know that the implications $$P \implies C_1\\ P \implies C_2\\ P \implies C_3$$ are true for some premise $P$ and some conditions $C_1, C_2$, and $C_3.$
If $C_1, C_2$, and $C_3$ are mutually exhaustive conditions, does it follow that $P$ is false?
MY ATTEMPT
I noticed that the three implications are logically equivalent to $$\left(P \implies C_1\right) \iff \left(\lnot P \lor C_1\right)\\ \left(P \implies C_2\right) \iff \left(\lnot P \lor C_2\right)\\ \left(P \implies C_3\right) \iff \left(\lnot P \lor C_3\right)$$ by Material Implication.
Hence, $$\left(\lnot P \lor C_1\right) \land \left(\lnot P \lor C_2\right) \land \left(\lnot P \lor C_3\right)$$ must be true.
Using Distribution of Disjunction over Conjunction, we obtain $$\Big(\left(\lnot P \lor C_1\right) \land \left(\lnot P \lor C_2\right) \land \left(\lnot P \lor C_3\right)\Big) \iff \Big(\lnot P \lor \left(C_1 \land C_2 \land C_3\right)\Big).$$
But the conditions $C_1, C_2$, and $C_3$ are mutually exhaustive. This means that $$C_1 \land C_2 \land C_3 \tag{*}$$ must be false.
We finally get that $\lnot P$ must be true, or in other words, $P$ must be false.
Is the proof argument in the section marked with a $(*)$ sound? I have some doubts since I am not sure if I should have used Distribution of Disjunction over Disjunction instead.
