How to give a formal proof for this particular case

Question

When making exercises for my Introduction to Logic course, I came across the following question which I can't seem to solve.

The question is:

Give a formal proof for ¬(A ∧ (¬A ∨ B)) ∨ B. Do not forget to provide justications.

I have tried many ways to prove this, but I think I miss something. I'm definitely not waiting for someone to make this exercise for me, but it would be amazing if someone could point me in the right direction.

Answer 1

$¬(A ∧ (¬A ∨ B)) ∨ B \iff ¬A ∨ ¬(¬A ∨ B) ∨ B \iff (¬A ∨ B) ∨ ¬(¬A ∨ B) $

define: $P = ¬A ∨ B$

so: $ (¬A ∨ B) ∨ ¬(¬A ∨ B) \iff P ∨ ¬P$ which is a tautology

-- proof that $P ∨ ¬P$ is a tautology --

let's evaluate: $P ∨ ¬P$

if $(P = T)$ then $(P ∨ ¬P) = (T ∨ F) = T$

if $(P = F)$ then $(P ∨ ¬P) = (F ∨ T) = T$