Help with discrete mathematics - inference and logical equivalence

Question

I'm doing textbook homework for discrete mathematics and I'm struggling to understand how to solve the following practice problem. I would understand if they gave me variables and asked me to use premises to construct a specific argument, but I don't understand how to reduce it just to a value of false? Any help or hints would be appreciated.

Question:

Using only the rules of inference and the logical equivalences, show that the following argument is a contradiction by reducing it to a value of "False". You may assume that all the premises given are true.

𝑎 → 𝑏

¬𝑏 ∧ 𝑐

¬𝑎 → 𝑑

𝑑 → ¬𝑒

𝑒 ∧ f

Answer 1

The implication $a \to b$ is equivalent to $\lnot a \lor b$

Similarly, the other two implications can be written as $a \lor d$ and $\lnot d \lor \lnot e$

The two conjunctions $\lnot b \land c$ and $e \land f$ can only be true, if all four operands are true: $\lnot b$, $c$, $e$ and $f$

This simplifies $\lnot a \lor b$ to $\lnot a$

$\lnot d \lor \lnot e$ is simplified to $\lnot d$ which in turn simplifies $a \lor d$ to $a$.

The last step unveils a contradiction, as $\lnot a$ was found to be true earlier on.