$(A \lor B) \to C$ and $(A \to C) \lor (B \to C)$ Which one entails the other?

Question

For a homework assignment I have to prove that one of the statements entails the other.

The statements are:

$(A \lor B) \to C$

$(A \to C) \lor (B \to C)$

The only thing that I got so far is either $\lnot(A \lor B) \to C$ or $(A \to C) \land (B \to C)$.

I can use the rules of Modus Ponens, Modus Tollens, Simplification, Conjunction, Disjunction, Conjunctive and Disjunctive Syllogism, Hypothethical Syllogism and Conditional Proof.

The equivalence rules I can use are Double Negation, De Morgan's Laws, Biconditional Equivalence,and I think, Transposition and Material Implication, too.

Is there someone who can help me? I've tried so much already...

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Edit:

The answer they wanted to hear was:

1. (A v B) -> C
2. A supp/CP
3. A v B disj
4. C 1,3MP
5. A -> C 2-4 CP
6. (A -> C) v (B -> C) 5 disj

Answer 1

HINT: Use the fact that $p\to q$ is equivalent to $\neg p\lor q$. Thus,

$$(A\lor B)\to C)\equiv\neg(A\lor B)\lor C\equiv(\neg A\land\neg B)\lor C\equiv(\neg A\lor C)\land(\neg B\lor C)\;,$$

where I’ve also used De Morgan’s law and distributivity. Now expand use the same fact to convert this to an expression involving $A\to C$ and $B\to C$, and compare that with $(A\to C)\lor(B\to C)$; one of the two expressions is easily shown to imply the other.