# a logical proposition

Here I have a proposition:

((¬p ∨ x) ∧ (p ∨ y)) → (x ∨ y)

To prove whether a tautology or contradiction or neither.

≡ ¬[((¬p ∨ x) ∧ (p ∨ y))] ∨ (x ∨ y) implication equivalence

≡ (¬(¬p ∨ x) ∨ ¬(p ∨ y)) ∨ (x ∨ y) DeMorgans

≡ ((p ∧ ¬x) ∨ (¬p ∧ ¬y)) ∨ (x ∨ y) DeMorgans

From here I'm not sure what to go on with. Have I made a mistake? Was that Distributive law an illegal move?

• You did make a mistake when applying the De Morgan's laws for the second time (you didn't change $\lor$ to $\land$). Sep 14, 2013 at 22:52

\begin{align} &\equiv (\lnot (\lnot p \lor x) \lor \lnot(p \lor y)) \lor (x \lor y) \tag{2}\\ \\ &\equiv (p \land \lnot x) \lor (\lnot p \land \lnot y) \lor (x \lor y)\\ \\ \tag{3}\end{align}
Now we can apply the distributive law to the first two parenthetical clauses, and then again in line $(5)$:
\begin{align} &\equiv [(p \lor (\lnot p \land \lnot y)) \land (\lnot x \lor (\lnot p \land \lnot y))] \lor (x \lor y)\tag{4} \\ \\ & \equiv [(p\lor \lnot p) \land (p \lor \lnot y) \land (\lnot x \lor \lnot p) \land (\lnot x \lor \lnot y)]\lor (x \lor y)\tag{5}\\ \\ \end{align}