Transformation rules 

Propositional calculus 
Rules of inference 
Rules of replacement 
Predicate logic 
In propositional logic, disjunction elimination^{[1]}^{[2]} (sometimes named proof by cases, case analysis, or or elimination), is the valid argument form and rule of inference that allows one to eliminate a disjunctive statement from a logical proof. It is the inference that if a statement implies a statement and a statement also implies , then if either or is true, then has to be true. The reasoning is simple: since at least one of the statements P and R is true, and since either of them would be sufficient to entail Q, Q is certainly true.
An example in English:
 If I'm inside, I have my wallet on me.
 If I'm outside, I have my wallet on me.
 It is true that either I'm inside or I'm outside.
 Therefore, I have my wallet on me.
It is the rule can be stated as:
where the rule is that whenever instances of "", and "" and "" appear on lines of a proof, "" can be placed on a subsequent line.
YouTube Encyclopedic

1/3Views:8 0256 4831 149

Disjunction Elimination example

Disjunction elimination and introduction

Question 8  Using Disjunction Elimination.mp4
Transcription
Formal notation
The disjunction elimination rule may be written in sequent notation:
where is a metalogical symbol meaning that is a syntactic consequence of , and and in some logical system;
and expressed as a truthfunctional tautology or theorem of propositional logic:
where , , and are propositions expressed in some formal system.
See also
References
 ^ "Archived copy". Archived from the original on 20150418. Retrieved 20150409.CS1 maint: archived copy as title (link)
 ^ http://www.cs.gsu.edu/~cscskp/Automata/proofs/node6.html