Question

I am having trouble with the following problem. I am unsure on how to even start.

Use propositional logic to prove that the following argument are valid:

(A⟶B) ⋀ (A⟶(B⟶C)) ⟶ (A⟶C)

Answer 1

$(A\to B) \wedge (A\to(B\to C)) \to(A\to C) \\ (A\to B) \wedge (A\to(B\to C))\\ (A'\vee B) \wedge(A \to(B' \vee C)) \text{Implication} \\ (A'\vee B) \wedge(A' \vee (B'\vee C)) \text{Implication}\\ (A'\vee B) \wedge (A'\vee B') \vee C \text{Assoiativity}\\ A'\vee(B \wedge B') \vee C \text{Distributive}\\ (A'\vee 0)\vee C \text{Known Contradiction}\\ A'\vee C \text{Absorbtion}\\ (A\to C)$

This shows that $(A\to B) \wedge (A\to(B\to C)) \to(A\to C)$. Hence the argument is valid.