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24 lines
1.2 KiB
24 lines
1.2 KiB
\setlength\subproofhorizspace{1.3em}
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\begin{logicproof}{2}
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\forall x \exists y \; \lnot (P(x) \land Q(y)) & \prem\\
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\exists y \lnot (P(x_0)\land Q(y)) & $\foralle1$\\
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\begin{subproof}
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\lnot (P(x_0)\land Q(y_0)) & $\assum$ $\freshVar{$y_0$}$\\
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\begin{subproof}
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\forall y (P(x_0)\land Q(y)) & $\assum$\\
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P(x_0)\land Q(y_0) & $\foralle4$\\
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\bot & $\nege3,5$
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\end{subproof}
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\lnot \forall y (P(x_0)\land Q(y)) & $\negi4-6$
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\end{subproof}
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\lnot \forall y (P(x_0)\land Q(y)) & $\existe2,3-7$\\
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\begin{subproof}
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\exists x \forall y (P(x)\land Q(y)) & $\assum$\\
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\begin{subproof}
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\forall y (P(x_0)\land Q(y)) & $\assum$ $\freshVar{$x_0$}$\\
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\bot & $\nege8,10$
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\end{subproof}
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\bot & $\existe9,10-11$
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\end{subproof}
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\lnot \exists x \forall y \; (P(x) \land Q(y)) & $\negi9-12$
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\end{logicproof}
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