Absorption (logic)
Transformation rules 

Propositional calculus 
Rules of inference 
Rules of replacement 

Predicate logic 
Absorption is a valid argument form and rule of inference of propositional logic.^{}^{} The rule states that if implies , then implies and . The rule makes it possible to introduce conjunctions to proofs. It is called the law of absorption because the term is "absorbed" by the term in the consequent.^{} The rule can be stated:
where the rule is that wherever an instance of "" appears on a line of a proof, "" can be placed on a subsequent line.
Formal notation
The absorption rule may be expressed as a sequent:
where is a metalogical symbol meaning that is a syntactic consequences of in some logical system;
and expressed as a truthfunctional tautology or theorem of propositional logic. The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:
where , and are propositions expressed in some formal system.
Examples
If it will rain, then I will wear my coat.
Therefore, if it will rain then it will rain and I will wear my coat.
Proof by truth table
T  T  T  T 
T  F  F  F 
F  T  T  T 
F  F  T  T 
Formal proof
Proposition  Derivation 

Given  
Material implication  
Law of Excluded Middle  
Conjunction  
Reverse Distribution  
Material implication 