a = a (equality is reflexive under Peano arithmetic) => a + c = a + c (addition is commutative under Peano arithmetic) => a + c = b + c (a == b) => a + c = b

a = a (equality is reflexive under Peano arithmetic) => a + c = a + c (addition is commutative under Peano arithmetic) => a + c = b + c (a == b) => a + c = b + d (c == d) qed