Proof of Theorem by Truth Table

Theorem
Let $$\phi$$ be a logical formula whose atoms are $$p_1, p_2, \ldots, p_n$$.

Let $$l$$ be the line number of any row in the truth table of $$\phi$$.

For all $$i: 1 \le i \ne n$$, let $$\hat {p_i}$$ be defined as:
 * $$\hat {p_i} = \begin{cases}

p_i & : \text {the entry in line } l \text { of } p_i \text { is } T \\ \neg p_i & : \text {the entry in line } l \text { of } p_i \text { is } F \end{cases}$$

Then:


 * $$\hat {p_1}, \hat {p_2}, \ldots, \hat {p_n} \vdash \phi$$ is provable if the entry for $$\phi$$ in line $$l$$ is $$T$$;
 * $$\hat {p_1}, \hat {p_2}, \ldots, \hat {p_n} \vdash \neg \phi$$ is provable if the entry for $$\phi$$ in line $$l$$ is $$F$$.

Proof

 * $$1:$$ Suppose $$\phi$$ is an atom $$p$$.

Then we need to show that $$p \vdash p$$ and $$\neg p \vdash \neg p$$.

These are proved in one line in the proof of the Law of Identity.


 * $$2:$$ Suppose $$\phi$$ is of the form $$\neg \phi_1$$.

There are two cases to consider:


 * Suppose $$\phi$$ evaluates to $$T$$.