183 Deductive Reasoning in First-order Logic 4.3.3 Natural Deduction for first-order logic with equality The system of Natural Deduction presented so far does not involve rules for the equality Such rules can be added to produce a sound and complete system of Natural Deduction for first-order logic with equality, as follows In each of the rules below, t, ti , si are terms (Ref) (ConFunc) s1 = t1 , , sn = tn f (s1 , , sn ) = f (t1 , , tn ) t=t (Sym) for every n-ary functional symbol f t1 = t2 t2 = t1 (ConRel) s1 = t1 , , sn = tn p(s1 , , sn ) → p(t1 , , tn ) (Tran) t1 = t2 , t2 = t3 t1 = t3 for every n-ary predicate symbol p Example 136 Using Natural Deduction with equality, derive ∀x∀y (f (x) = y → g (y ) = x) ND ∀z (g(f (z )) = z ) where f, g are unary function symbols ∀x∀y (f (x) = y → g(y ) = x) ∀y (f (x) = y → g (y ) = x) (∀E ) f (x) = f (x) f (x) = f (x) → g (f (x)) = x (→ E ) g (f (x)) = x (∀I ) ∀z (g(f (z )) = z ) (∀E ) Finally, two important general results are as follows Theorem 137 [Equivalent replacement] For any formula A(x) and terms s, t free for x in A, the following is derivable in ND: s=t ND A[s/x] ↔ A[t/x] The proof can be done by induction on A and is left as an exercise The proof of the following fundamental result, for ND with equality, will be outlined in Section 4.6