Recently Beyersdorff, Bonacina, and Chew (ITCS'16) introduced a natural class of Frege systems for quantified Boolean formulas (QBF) and showed strong lower bounds for restricted versions of these systems. Here we provide a comprehensive analysis of the new extended Frege system from Beyersdorff et al., denoted EF+$\forall$red, which is a natural extension of classical extended Frege EF.

Our main results are the following: Firstly, we prove that the standard Gentzen-style system $G^*_1$ p-simulates EF+$\forall$red and that $G^*_1$ is strictly stronger under standard complexity-theoretic hardness assumptions.

Secondly, we show a correspondence of EF+$\forall$red to bounded arithmetic: EF+$\forall$red can be seen as the non-uniform propositional version of intuitionistic $S^1_2$. Specifically, intuitionistic $S^1_2$ proofs of arbitrary statements in prenex form translate to polynomial-size EF+$\forall$red proofs, and EF+$\forall$red is in a sense the weakest system with this property.

Finally, we show that unconditional lower bounds for EF+$\forall$red would imply either a major breakthrough in circuit complexity or in classical proof complexity, and in fact the converse implications hold as well. Therefore, the system EF+$\forall$red naturally unites the central problems from circuit and proof complexity.

Technically, our results rest on a formalised strategy extraction theorem for EF+$\forall$red akin to witnessing in intuitionistic $S^1_2$ and a normal form for EF+$\forall$red proofs.