Substitutivity of Equivalence Let A,M and N be wffs and let AMN be the result of replacing M by N at zero or more occurrences (henceforth called designate occurrences) of M in A. 1. AMN is a wff. 2. If |= M ≡ N then |= A ≡ AMN
Some Properties 1. ∆n is consistent. 2. Γ ⊆ ∆n ⊆ ∆n+1 ⊆ ∆Γ 3. ∆Γ is complete. 4. If ∆Γ ` A then there exists n ∈ N such that ∆n ` A. 5. A ∈ ∆Γ iff ∆Γ ` A 6. ∆Γ is consistent
Theory of Equivalence Relations (A, R) (E1) For all x : xRx. (E2) For all x, y : If xRy then yRx. (E3) For all x, y, z : If xRy and yRz then xRz. Logic in Computer Science – p.2/16
