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27 lines
1.4 KiB
27 lines
1.4 KiB
\setlength\subproofhorizspace{1.3em}
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\begin{logicproof}{4}
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\lnot \exists x \forall y \; (P(x) \land Q(y)) & \prem\\
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\begin{subproof}
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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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\exists x \forall y (P(x)\land Q(y)) & $\existi2$\\
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\bot & $\nege1,3$
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\end{subproof}
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\lnot \forall y (P(x_0)\land Q(y)) & $\negi2-4$\\
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\begin{subproof}
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\lnot \exists y \lnot (P(x_0)\land Q(y)) & $\assum$\\
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\begin{subproof}
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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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\exists y \lnot (P(x_0) \land Q(y)) & $\existi7$\\
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\bot & $\nege6,8$
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\end{subproof}
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P(x_0) \land Q(y_0) & $PBC 7-9$
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\end{subproof}
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\forall y (P(x_0)\land Q(y)) & $\foralli7-10$\\
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\bot & $\nege5,11$
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\end{subproof}
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\exists y \lnot (P(x_0)\land Q(y)) & $PBC 6-12$
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\end{subproof}
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\forall x \exists y \; \lnot (P(x) \land Q(y)) & $\foralli2-13$
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\end{logicproof}
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