Rule of Substitution

Theorem
Let $S$ be a sequent that has been proved.

Then a proof can be found for any substitution instance of $S$.

Proof
This is apparent from inspection of the proof rules themselves.

The rules concern only the broad structure of the propositional formulas involved, and this structure is unaffected by substitution.

By performing the substitutions systematically throughout the given sequent, all applications of proof rules remain correct applications in the sequent.

Also known as
Some sources amplify the name to rule of uniform substitution.

Applications
This proof leads on to the Rule of Sequent Introduction.

Technical Note
When invoking the Rule of Substitution in a tableau proof, use the Substitution template:



where:
 * is the number of the line on the tableau proof where the Rule of Substitution is to be invoked
 * is the pool of assumptions (comma-separated list) of the statement on which the Rule of Substitution is to be used
 * is the statement of logic that is to be displayed in the Formula column, without the  delimiters
 * is the line of the tableau proof of the statement on which the Rule of Substitution is to be used
 * is the list of instances which are being substituted that will be displayed in the Notes column.