The proof appeals to the Principle of Structural Induction, applied to the statement: :If $\mathbf C$ occurs along $\Gamma$, then $v \models_{\mathrm{BI}} \mathbf C$. When $\mathbf C$ is basic, the result holds per assumption. Suppose $\mathbf C$ is not basic. It is seen that one of the propositional tableau construction rules applies to $\mathbf C$. If $\mathbf C = \neg\neg\mathbf A$, then it can only be used by the $\boxed{\neg\neg}$ rule. Hence, $\mathbf A \in \Phi \left[{\Gamma}\right]$, and by induction hypothesis, $v \models_{\mathrm{BI}} \mathbf A$. By Double Negation, $v \models_{\mathrm{BI}} \neg\neg\mathbf A$. If $\mathbf C$ is $\mathbf A \land \mathbf B$, then it can only be used by the $\boxed\land$ rule. Therefore, $\mathbf A, \mathbf B \in \Phi \left[{\Gamma}\right]$, and by induction hypothesis: :$v \models_{\mathrm{BI}} \mathbf A,\mathbf B$ By the truth function for $\land$, it follows that: :$v \models_{\mathrm{BI}} \mathbf A \land \mathbf B$ If $\mathbf C$ is $\neg \left({\mathbf A \land \mathbf B}\right)$, then it can only be used by the $\boxed{\neg\land}$ rule. Therefore, $\neg\mathbf A \in \Phi \left[{\Gamma}\right]$ or $\neg\mathbf B \in \Phi \left[{\Gamma}\right]$. By induction hypothesis, it follows that: :$v \models_{\mathrm{BI}} \neg\mathbf A$ or $v \models_{\mathrm{BI}} \neg\mathbf B$ By the truth function for $\land$, it follows that: :$v \models_{\mathrm{BI}} \neg \left({\mathbf A \land \mathbf B}\right)$ If $\mathbf C$ is $\mathbf A \lor \mathbf B$, then it can only be used by the $\boxed\lor$ rule. Therefore, $\mathbf A \in \Phi \left[{\Gamma}\right]$ or $\mathbf B \in \Phi \left[{\Gamma}\right]$. By induction hypothesis, it follows that: :$v \models_{\mathrm{BI}} \mathbf A$ or $v \models_{\mathrm{BI}} \mathbf B$ By the truth function for $\lor$, it follows that: :$v \models_{\mathrm{BI}} \mathbf A \lor \mathbf B$ If $\mathbf C$ is $\neg \left({\mathbf A \lor \mathbf B}\right)$, then it can only be used by the $\boxed{\neg\lor}$ rule. Therefore, $\neg\mathbf A, \neg\mathbf B \in \Phi \left[{\Gamma}\right]$, and by induction hypothesis: :$v \models_{\mathrm{BI}} \neg\mathbf A,\neg\mathbf B$ By the truth function for $\lor$, it follows that: :$v \models_{\mathrm{BI}} \neg \left({\mathbf A \lor \mathbf B}\right)$ If $\mathbf C$ is $\mathbf A \implies \mathbf B$, then it can only be used by the $\boxed\implies$ rule. Therefore, $\neg\mathbf A \in \Phi \left[{\Gamma}\right]$ or $\mathbf B \in \Phi \left[{\Gamma}\right]$. By induction hypothesis, it follows that: :$v \models_{\mathrm{BI}} \neg\mathbf A$ or $v \models_{\mathrm{BI}} \mathbf B$ By the truth function for $\implies$, it follows that: :$v \models_{\mathrm{BI}} \mathbf A \implies \mathbf B$ If $\mathbf C$ is $\neg \left({\mathbf A \implies \mathbf B}\right)$, then it can only be used by the $\boxed{\neg\implies}$ rule. Therefore, $\mathbf A, \neg\mathbf B \in \Phi \left[{\Gamma}\right]$, and by induction hypothesis: :$v \models_{\mathrm{BI}} \mathbf A,\neg\mathbf B$ By the truth function for $\implies$, it follows that: :$v \models_{\mathrm{BI}} \neg \left({\mathbf A \implies \mathbf B}\right)$ If $\mathbf C$ is $\mathbf A \iff \mathbf B$, then it can only be used by the $\boxed\iff$ rule. Therefore, $\mathbf A\land \mathbf B \in \Phi \left[{\Gamma}\right]$ or $\neg\mathbf A \land \neg\mathbf B \in \Phi \left[{\Gamma}\right]$. By induction hypothesis, it follows that: :$v \models_{\mathrm{BI}} \mathbf A,\mathbf B$ or $v \models_{\mathrm{BI}} \neg\mathbf A,\neg\mathbf B$ By the truth function for $\iff$, it follows that: :$v \models_{\mathrm{BI}} \mathbf A \iff \mathbf B$ If $\mathbf C$ is $\neg \left({\mathbf A \iff \mathbf B}\right)$, then it can only be used by the $\boxed{\neg\iff}$ rule. Therefore, $\mathbf A\land \neg\mathbf B \in \Phi \left[{\Gamma}\right]$ or $\neg\mathbf A \land \mathbf B \in \Phi \left[{\Gamma}\right]$. By induction hypothesis, it follows that: :$v \models_{\mathrm{BI}} \mathbf A,\neg\mathbf B$ or $v \models_{\mathrm{BI}} \neg\mathbf A,\mathbf B$ By the truth function for $\iff$, it follows that: :$v \models_{\mathrm{BI}} \neg \left({\mathbf A \iff \mathbf B}\right)$ Having dealt with all cases, it follows from the Principle of Structural Induction that: :$v \models_{\mathrm{BI}} \mathbf C$ for all WFFs $\mathbf C$ that occur along $\Gamma$ That is to say: :$v \models_{\mathrm{BI}} \Phi \left[{\Gamma}\right]$ {{qed}} Category:Propositional Tableaus da9sr6r7oljqe5xoif0lui89r0445rn