142 Logic as a Tool Theorem 116 (Equivalent replacement) For any formula A(x) and terms s, t free for x in A, the following holds: |= s = t → A[s/x] ↔ A[t/x] The proof is by structural induction on the formula A(x), and is left as an exercise for the reader 3.4.5 Logical equivalence in first-order logic Logical equivalence in first-order logic is based on the same idea as in propositional logic: the first-order formulae A and B are logically equivalent if always one of them is true if and only if the other is true This is defined formally as follows Definition 117 The first-order formulae A and B are logically equivalent, denoted A ≡ B , if for every structure S and variable assignment v in S: S , v |= A if and only if S , v |= B In particular, if A and B are sentences, A ≡ B means that every model of A is a model of B , and every model of B is a model of A The following theorem summarizes some basic properties of logical equivalence Theorem 118 A ≡ A If A ≡ B then B ≡ A If A ≡ B and B ≡ C then A ≡ C If A ≡ B then ¬A ≡ ¬B , ∀xA ≡ ∀xB , and ∃xA ≡ ∃xB If A1 ≡ B1 and A2 ≡ B2 then A1 ◦ A2 ≡ B1 ◦ B2 where ◦ is any of ∧, ∨, →, ↔ The following are equivalent: (a) A ≡ B ; (b) |= A ↔ B ; and (c) A |= B and B |= A Theorem 119 The result of renaming of any variable in any formula A is logically equivalent to A Consequently, every formula can be transformed into a logically equivalent clean formula Example 120 Some examples of logical equivalences between first-order formulae are as follows • Any first-order instance of a pair of equivalent propositional formulae is a pair of logically equivalent first-order formulae For example: ¬¬∃xQ(x, y ) ≡ ∃xQ(x, y ) (being an instance of ¬¬p ≡ p); ∃xP (x) → Q(x, y ) ≡ ¬∃xP (x) ∨ Q(x, y ) (being an instance of p → q ≡ ¬p ∨ q )