How to derive $¬X \lor ¬Y \lor ¬Y$ from $Z ⊃ (¬X \lor ¬Y)$ and $¬Z ⊃ ¬ Y$

Question

I know that one can derive $¬X \lor ¬Y \lor ¬Y$ (which simplifies to $¬X \lor ¬Y$, right?) from $Z ⊃ (¬X \lor ¬Y)$ and $¬Z ⊃ ¬ Y$ but I don't know how you do this.

Maybe one just has to use constructive dilemma?

Answer 1

Yes, you can use $¬Z \lor Z$ and Constructive dilemma.

Or you can use $¬Z \lor Z$ and Proof by cases :

1) $¬Z \lor Z$

2) $Z⊃(¬X∨¬Y)$

3) $¬Z⊃¬Y$

4) $¬Y⊃(¬X∨¬Y)$ --- Disjunction introduction

5) $¬Z⊃(¬X∨¬Y)$ --- from 3) and 4) by Hypothetical syllogism

6) $¬X \lor ¬Y$ --- from 1) 2) and 5) by Disjunction elimination.

And then again Disjunction introduction to get: $(¬X∨¬Y)∨¬Y$

Answer 2

If you have Hypothetical Syllogism and Contraposition, you can do it without having to assume $Z \lor \neg Z$:

$1. Z \rightarrow (\neg X \lor \neg Y) \quad Premise$

$2. \neg Z \rightarrow \neg Y \quad Premise$

$3. Y \rightarrow Z \quad Contraposition \ 2$

$4. Y \rightarrow (\neg X \lor \neg Y) \quad Hypothetical \ Syllogism \ 1,3$

$5. \neg Y \lor \neg X \lor \neg Y \quad Implication \ 4$

$6. \neg X \lor \neg Y \lor \neg Y \quad Commutation \ 5$