Show, by equivalence, that: A ⇒ (B ⇒ C) ≡ (A ⇒ B) ⇒ (A ⇒ C)

Question

Sorry for my previous mistake as I have included my own attempts in this one. I am trying learn LaTex but it will take some time so forgive me if use normal format at this question:

(A ⇒ B) ⇒ (A ⇒ C) = (~~A ∨ ~B) ∨ (~A ∨ C) Conditional Law

              = (A ∨ ~B) ∨ (~A ∨ C)         Double Negation
              = A ∨ ~A ∨ ~B ∨ C
              = T ∨ (~B ∨ C)                Tautology


  A ⇒ (B ⇒ C) = A ⇒ (~B ∨ C)                Conditional Law
              = ~A ∨ (~B ∨ C)               Conditional Law

I'm stuck at this part and can't think solve the rest of it.

Please let me know if this question is acceptable by the website users.

Thank you all.

Answer 1

You made a simple mistake in the first expression:

$$(A\Rightarrow B)\Rightarrow (A\Rightarrow C)$$$$\equiv \neg(\neg A\vee B)\vee(\neg A\vee C)$$$$\equiv (A\wedge\neg B)\vee(\neg A\vee C)$$$$\equiv ((A\wedge \neg B)\vee \neg A)\vee C$$$$\equiv (\neg A\vee \neg B)\vee C$$