Using properties of semantic equivalence I would like to prove:
(p⇒r) ∧ (q⇒r) ≡ (p V q) ⇒ r
I have wound up with the correct result however the steps it took to get there seemed excessive. If anyone can tell me if I have added unnecessary steps it would be very helpful. I am asked to not use a rule twice in one step - hence the separation of implication rules in the beginning. Thank you in advance.
(implication) (﹁p) V r ∧ (q ⇒ r) (implication) (﹁p) V r ∧ (﹁q) V r (associativity) (﹁p V r) ∧ (﹁q V r) (commutavity) (r V ﹁p) ∧ (r V ﹁q) (distributivity) r V (﹁p ∧ ﹁q) (commutativity) (﹁p ∧ ﹁q) V r (deMorgan's) ﹁(p ∧ q) V r (implication) (p ∧ q) ⇒ r