A quantified Boolean formula is true if and only if the existential player has a winning strategy in its game

If $\mathrm{\Phi }$ has no quantifiers, then [L1] says that $\mathrm{\Phi }$ is true exactly when its quantifier-free matrix is true, and [L2] says that this is exactly the condition for the existential player to win the zero-move game.

Suppose $\mathrm{\Phi }=\mathrm{\exists }x\mathrm{\Psi }(x)$. By [L1], $\mathrm{\Phi }$ is true if and only if at least one of $\mathrm{\Psi }(0)$ or $\mathrm{\Psi }(1)$ is true. By the induction hypothesis, that happens exactly when the existential player has a winning strategy for at least one of the two suffix games, and then the existential player wins the whole game by choosing that first move. Conversely, any winning strategy for the whole game determines such a first move.

Suppose instead $\mathrm{\Phi }=\mathrm{\forall }x\mathrm{\Psi }(x)$. By [L1], $\mathrm{\Phi }$ is true if and only if both $\mathrm{\Psi }(0)$ and $\mathrm{\Psi }(1)$ are true. By the induction hypothesis, that is equivalent to the existential player having a winning strategy after either first move of the universal player, which is exactly what [L2] means for the whole game.

Steps 1.1, 1.2, and 1.3 prove both directions of the equivalence.