Rule of Substitution

Context
This proof is demonstrated in the field of natural deduction, but is broadly applicable in other fields of mathematics.

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 statement forms 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.

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