Tautology

A tautology (otherwise known as valid) is a formula that is true under all [[Interpretation|interpretations]], every row in a truth table of a tautology makes it true.

We can denote a tautology by $\boxed{\models x}$, where $x$ is a formula which is a tautology.

$p$	$\neg p$	$p\vee \neg q$
T	F	T
F	T	T

Given the set of formulas, $\Delta = \{ A_1, A_2, A_3 \}$, provided $\Delta \models \varphi$:
 
$$
	\begin{aligned}
		\Delta &\models \varphi \\
		A_1, A_2, A_3 &\models \varphi \\
		A_1 \land A_2 \land A_3 &\models \varphi \\
		A_1 , A_2 &\models A_3 \rightarrow \varphi \\
		A_1 &\models A_2 \rightarrow (A_3 \rightarrow \varphi) \\
		\emptyset &\models A_1 \rightarrow (A_2 \rightarrow (A_3 \rightarrow \varphi)) \\
	\end{aligned}
$$