Rule of Replacement ·Assume中(A)is a proposition formula (B)is another proposition formula containing a logical expression A obtained from (A)by replacing all occurrences of A with B IfB=A,thenΦ(A)≡Φ(B) (DeMorgan's laws)-(PVQ)=(-PA-Q): To prove(pVq)→r≡(p→r)A(q→r) (PAQ)三（一PV一Q mplication and(P-→Q)三(-PVQ: (p→r)Λ(q→r) its negation) (P一Q)三(PA-Q: (Double negation)(P）≡P. ≡(pVr)∧(qVr) Implication (Distributive property)(PA(QVR))=((PAQ)V(PAR)); (PV(OAR))=((PVO)A(PVR)): ≡(pAq)Vr Distribution (Associative property) (PA(QAR)三(PAQ)AR: (PV(QVR)≡(PVQ)VR): ≡(pVq)Vr DeMorgan's law (Commutative property）(PAQ)三(QAP: (PVQ)≡(QVP) ≡(pVq)→r ImplicationRule of Replacement • Assume 𝛷(𝐴) is a proposition formula containing a logical expression 𝐴 If 𝑩 ≡ 𝑨, then 𝜱(𝑨) ≡ 𝜱(𝑩) • 𝛷(𝐵) is another proposition formula obtained from 𝛷(𝐴) by replacing all occurrences of 𝐴 with 𝐵 To prove 𝒑 ∨ 𝒒 → 𝒓 ≡ 𝒑 → 𝒓 ∧ 𝒒 → 𝒓 𝒑 → 𝒓 ∧ 𝒒 → 𝒓 ≡ ¬𝒑 ∨ 𝒓 ∧ ¬𝒒 ∨ 𝒓 Implication ≡ ¬𝒑 ∧ ¬𝒒 ∨ 𝒓 Distribution ≡ ¬ 𝒑 ∨ 𝒒 ∨ 𝒓 DeMorgan’s law ≡ 𝒑 ∨ 𝒒 → 𝒓 Implication ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡