Proof by induction


THEOREM   
$$\begin{align}
       &\forall x \in \mathbb{N}: P(x, 0), \\
       &\forall y \in \mathbb{N}: P(0, \Suc(y)), \\
       &\forall x,y \in \mathbb{N} : P(x, y) \Rightarrow P(\Suc(x), \Suc(y)) \\
\vdash &\forall x,y \in \mathbb{N}: P(x, y)
\end{align}$$

Proof available