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: \begin{align}

(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}