Geodesic
A~normal form theorem for second-order classical logic with an axiom of choice
Izvestiya. Mathematics , Tome 32 (1989) no. 3, pp. 587-605.

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A cut-elimination theorem for the second-order logic with an axiom of choice of type $0,1$ or $1,1$ is proved. In the first case the Päppinghaus scheme is applied; in the second the calculus with an epsilon-symbol for predicates is used. Bibliography: 5 titles. 
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G. E. Mints. A~normal form theorem for second-order classical logic with an axiom of choice. Izvestiya. Mathematics , Tome 32 (1989) no. 3, pp. 587-605. http://geodesic.mathdoc.fr/item/IM2_1989_32_3_a6/

[1] Tait W. W., “A non-constructive proof of Gentzen's Hauptsatz for second order predicate logic”, Bull. Amer. Math. Soc., 72 (1966), 980–983 | DOI | MR | Zbl

[2] Päppinghaus P., “Completeness properties of classical theories of finite type and thenormal form theorem”, Disc. Math., 207 (1983) | MR | Zbl

[3] Girard I. J., “Three-valued logic and cut-elimination”, Disc. Math., 136 (1976) | MR | Zbl

[4] Kreisel G., Takeuti G., “Formally self-referential propositions for cut free classical analysis and related systems”, Disc. Math., 118 (1974) | MR | Zbl

[5] Schütte K., “Syntactical and semantical properties of simple type theory”, J. Symbol Log., 25 (1960), 305–326 | DOI | MR | Zbl