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

Question

((A ∧ (A ⇒ B)) ⇒ B) ≡ T

(~(A ∧ (A ⇒ B)) ∨ B)..............Conditional Law

(~(A ∧ (~A ∨ B)) ∨ B).............Conditional Law

((~A ∨ ~(~A ∨ B)) ∨ B)............De Morgan's Law

((~A ∨ (A ∧ ~B)) ∨ B).............De Morgan's Law

I'm stuck here and don't know what law should I apply to go further, I tried distributive law but I still end with ~A at the end.

Answer 1

I would say

(~A ∨ (A ∧ ~B) ∨ B) ≡ ((~A ∨ A) ∧ (~A ∨ ~B) ∨ B) ≡

(~A ∨ ~B ∨ B) ≡ (~B ∨ B) ≡ T