In past lessons, we did proofs of various rules of replacement. In Introducing Conditional Proof, we did proofs of Tautology, then as homework, proofs of Transposition and the conjunctive forms of Commutation and Association. In Rules of Inference for Biconditionals, we did proofs of Material Equivalence. In Nesting Conditional Proofs, we did a proof of Exportation. And in Proving Disjunctions with Conditional Proof, we did proofs of the disjunctive forms of Commutation and Association. In this lesson, we are doing proofs of the rule of Distribution.

We have three rules of replacement left to prove. These are Double Negation, De Morgan's, and Material Implication. What these share in common is the appearance of the negation operator, which we have been representing with a tilde. One of these can be proven with just the rules of inference, the rules of replacement, and conditional proof. The others, as far as I can tell, cannot. Before the next lesson, look into which one can be proven and try to prove it. In the next lesson, we will be learning about a method for proving the negation of our assumption, and we will use it in proofs for the rules of replacement we can't yet prove.

For Next Time, try to prove:
Double Negation:
P ≡ ~~P
Material Implication:
(~P ∨ Q) ≡ (P ⊃ Q)
De Morgan's:
~(P & Q) ≡ (~P ∨ ~Q)
~(P ∨ Q) ≡ (~P & ~Q)

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