Rule of Substitution

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

Then we may infer any sequent $S'$ resulting from $S$ by substitutions for letters.

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 of letters.

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.