Show that α_1 -> ~α_4 can be derived from the given premises: (α_1-> (α_2 ∨α_3)), α_2 -> ~α_1, α_4 -> ~α_3

 Step 1: (α_1-> (α_2 ∨α_3)) (Premise)

Step 2: α_1(Assumed Premise)

Step 3: α_2 ∨α_3 (Modus Ponens, Step 1, 2)

Step 4: α_2 -> ~α_1 (Premise)

Step 5: ~α_2 (Modus Tollens, Step 2, step 4)

Step 6: α_3 (Disjunctive Syllogism, Step 1, 2, 4)

Step 7: α_4-> ~α_3 (Premise)

Step 8: ~α_4 (Modus Tollens, Step 1, 2, 4, 7)

Step 9: α_1 -> ~α_4 (Step 2, 4)

Hence proved