Use equational proofs to solve problems (Logic)

Question

Use equational proofs to solve the problem:

$ \vdash A \lor B \equiv A \lor \lnot B \equiv A $

These are the Axioms and the theorems :

Answer 1

Start with the following instance of (2.4.12):

$$\vdash \lnot B \lor A \equiv B \lor A \equiv A.$$

By the symmetry of equivalence (2) we obtain:

$$\vdash B \lor A \equiv \lnot B \lor A \equiv A.$$

By the symmetry of disjunction (6) we obtain:

$$\vdash A \lor B \equiv A \lor \lnot B \equiv A.$$

I'm not familiar with the system; I apologize if this is not the sort of derivation you're looking for.