Deduction Theorem (Propositional Logic)

Let and Then if and only if Syntactic Entailment (Propositional Logic)

Proof

Assume . Write down the proof of from and add the following lines to obtain a proof of from :

Now let’s assume . Let be a proof of from . We prove that by induction on

1. If is an axiom or . Then

2. If . In this case since
3. If s.t. By induction we can write down proofs of and from . We add the lines:

(p \implies(t_{j}\implies t_{i}))\implies((p \implies t_{j})\implies(p \implies t_{i}))\quad %quad \quad %quad & \text{(A2)} \ (p \implies t_{j}) \implies( p \implies t_{i})\quad %quad \quad %quad & \text{(MP)} \ p \implies t_{i}\quad %quad \quad %quad & \text{(MP)} \end{align}