Mathematical Proof and the Principles of Mathematics/Logic/Rules of inference summary

This is a list of the rules of inference given in previous sections. In the notation being used, a solid horizontal bar means that the statement below is a valid deduction from the statement(s) above. A vertical bar with a horizontal bar connected to it means that whatever is to the right of the vertical bar is a subproof, and whatever is above the horizontal bar are assumption(s) and whetever is below the horizontal bar is what has been derived. The names given are just placeholders and no guarantees are made that they are standard in any way. Propositional logic Rules not requiring subproofs Iteration Use of contradiction Disjunction by first case or Disjunction by second case or First use of conjunction and Second use of conjunction and Implication from the conclusion implies Implication from false assumption not implies Double negation not not Equivalence to implication iff implies Equivalence to converse iff implies Conjunction by components and Use of disjunction, first alternative false or not Use of disjunction, second alternative false or not Use of implication, from premise (modus ponens) implies Use of implication, from false conclusion (modus tollens) implies not not Rules requiring one subproof Implication by direct proof implies