in five different ways:
(a) Prove it using the method of Section 4.1. That is, either transform the antecedant into the consequent or vice versa. (The consequent has "more structure", so it may be better to start with it.) If you transform antecedant into consequent, then in each step the relationship between the two relevant expressions expressions must be either ≡ or ⇒ (implies). If you go in the opposite direction, the relationship in each step must be either ≡ or ⇐ (is consequence of)).
(b) Prove it using the method of assuming the antecedant.
(c) Prove it using the method of proof by case analysis. Use p (i.e., E[p := true]) and ¬p (i.e., E[p := false]) as the two cases. (In other words, prove each of ((p ∧ q) ⇒ (p ∧ r ⇒ q))[p:=true] and ((p ∧ q) ⇒ (p ∧ r ⇒ q))[p:=false].)
(d) Prove it using the method of proof by contradiction. (To prove A by contradiction, show ¬A ⇒ false.)
(e) Prove it using the method of proof by contrapositive. (To prove A ⇒ B by contrapositive, show ¬B ⇒ ¬A.)
2. Using the methods of Gries & Schneider, prove the validity of the following argument.
Spock is brilliant. If Vulcans are logical and Spock is brilliant, then Kirk is arrogant. If Vulcans are logical and Scotty got drunk last night and Kirk is arrogant, then Sulu is a good helmsman. If Scotty got drunk last night, then Vulcans are logical. If Spock is brilliant, then Scotty got drunk last night. Therefore, Sulu is a good helmsman.