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