Formal logic proof using 19 rules of inference

Question

I am working on a problem in logic. The proof is supposed to involve only 19 rules of inference.

The problem is:

Given:

$$ Z \implies A $$ $$ Z \lor A $$

Deduce:

$$ A$$

My proof is

$$ (Z \implies A) \land (Z \lor A) $$ $$ (Z \implies A) \land (\lnot \lnot Z \lor A) \text{(Double negation)}$$ $$ (Z \implies A) \land (\lnot Z \implies A) \text {(implication)} $$ $$ Z \lor \lnot Z \text{(tautology)} $$ $$ A \lor A \text {(constructive dilemma)} $$ $$ A \text{(tautology)} $$

What is my mistake?

Answer 1

I think that it would be much better illustrated as follows:

$ (Z \implies A) \land (Z \lor A) $

$ (\lnot Z \lor A) \land (Z \lor A) $

$ (A \lor \lnot Z) \land (A \lor Z) $

$ A \land (\lnot Z \lor Z) $

$ A \land true $

$ A $