Here are some commonly used inference rules

In Propositional Logic

Conditional

  • implication introduction

(not to be confused with conditional proofs)

  • implication elimination (modus ponens, affirming the antecedent): “If X implies Y, and X is true, then Y is true”
  • denying the consequent (modus tollens, denying the consequent): “If X implies Y, and Y is false, then X is false”

Biconditional

  • biconditional introduction
  • biconditional elimination

and

Conjunction and Disjunction

  • conjunction introduction
  • conjunction elimination and

  • disjunction introduction