Bentley used the invalid proof to introduce his students to the concept of mathematical proofs. Such explanations may include diagrams and formal mathematical proofs. This page lists notable examples of incomplete published mathematical proofs. Nonetheless, these laws are helpful in making valid inferences in proofs and deductive arguments. However, photocopies of the proofs circulated among scholars. The rules allow one to transpose propositional variables within logical expressions in logical proofs. See Noether's theorem for proofs and details. It improves the rigor of certain existence and equivalence proofs. There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. This was a key step in their first proofs of the prime number theorem. Therefore, it is preferred in proofs and theoretical arguments. Unique factorization is essential to many proofs of number theory. The rules allow one to move parentheses in logical expressions in logical proofs. Flew also criticised several of the other philosophical proofs for God's existence.