Disjunction elimination

In <a href="/facts/Propositional_logic/MNt5v31V">propositional logic</a>, disjunction elimination (sometimes named proof by cases, case analysis, or or elimination) is the <a href="/facts/Validity_(logic)/oMGkywGC">valid</a> <a href="/facts/Argument_form/foVBtMX4">argument form</a> and <a href="/facts/Rule_of_inference/QV5NklQ9">rule of inference</a> that allows one to eliminate a <a href="/facts/Logical_disjunction/jbI2EGlM">disjunctive statement</a> from a <a href="/facts/Formal_proof/6nEhV4cw">logical proof</a>. It is the <a href="/facts/Inference/RBbe95bP">inference</a> that if a statement 
 
 
 
 P
 
 
 {\displaystyle P}
 
 implies a statement 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 and a statement 
 
 
 
 R
 
 
 {\displaystyle R}
 
 also implies 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
, then if either 
 
 
 
 P
 
 
 {\displaystyle P}
 
 or 
 
 
 
 R
 
 
 {\displaystyle R}
 
 is true, then 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 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 <a href="/facts/English_language/tpkxGt7b">English</a>:

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:

P
              →
              Q
              ,
              R
              →
              Q
              ,
              P
              ∨
              R
            
            
              ∴
              Q
            
          
        
      
    
    {\displaystyle {\frac {P\to Q,R\to Q,P\lor R}{\therefore Q}}}

where the rule is that whenever instances of "
 
 
 
 P
 →
 Q
 
 
 {\displaystyle P\to Q}
 
", and "
 
 
 
 R
 →
 Q
 
 
 {\displaystyle R\to Q}
 
" and "
 
 
 
 P
 ∨
 R
 
 
 {\displaystyle P\lor R}
 
" appear on lines of a proof, "
 
 
 
 Q
 
 
 {\displaystyle Q}
 
" can be placed on a subsequent line.

Disjunction elimination open-in-new

Disjunction elimination