User:The Cube Root Of Infinity/sandbox

$$ \begin{array}{rl} \text{Ax. 1.} & \left(P(\varphi) \;\wedge\; \Box \; \forall x (\varphi(x) \Rightarrow \psi(x))\right) \;\Rightarrow\; P(\psi) \\ \end{array} $$

$$ \begin{array}{rl}

\text{Ax. 2.} & P(\neg \varphi) \;\Leftrightarrow\; \neg P(\varphi) \\

\end{array} $$ $$ \begin{array}{rl}

\text{Th. 1.} & P(\varphi) \;\Rightarrow\; \Diamond \; \exists x \; \varphi(x) \\

\end{array} $$ $$ \begin{array}{rl} \text{Df. 1.} & G(x) \;\Leftrightarrow\; \forall \varphi (P(\varphi) \Rightarrow \varphi(x)) \\

\end{array} $$ $$ \begin{array}{rl} \text{Ax. 3.} & P(G) \\

\end{array} $$ $$ \begin{array}{rl} \text{Th. 2.} & \Diamond \; \exists x \; G(x) \\

\end{array} $$ $$ \begin{array}{rl} \text{Df. 2.} & \varphi \text{ ess } x \;\Leftrightarrow\; \varphi(x) \wedge \forall \psi \left(\psi(x) \Rightarrow \Box \; \forall y (\varphi(y) \Rightarrow \psi(y))\right) \\

\end{array} $$ $$ \begin{array}{rl} \text{Ax. 4.} & P(\varphi) \;\Rightarrow\; \Box \; P(\varphi) \\

\end{array} $$ $$ \begin{array}{rl} \text{Th. 3.} & G(x) \;\Rightarrow\; G \text{ ess } x \\

\end{array} $$ $$ \begin{array}{rl} \text{Df. 3.} & E(x) \;\Leftrightarrow\; \forall \varphi (\varphi \text{ ess } x \Rightarrow \Box \; \exists y \; \varphi(y)) \\

\end{array} $$ $$ \begin{array}{rl} \text{Ax. 5.} & P(E) \\

\end{array} $$ $$ \begin{array}{rl} \text{Th. 4.} & \Box \; \exists x \; G(x)

\end{array} $$