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26 lines
617 B
26 lines
617 B
This sequent is provable.
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\begin{logicproof}{2}
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(p \land q) \imp (p \land r) & \prem\\
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q & \prem\\
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\begin{subproof}
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p & \assum \\
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p \land q & $\andi 3, 2$\\
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p \land r & $\impe 4, 1$\\
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r & $\ande{2} 6$
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\end{subproof}
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p \imp r & $\impi 3-6$\\
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\begin{subproof}
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\neg(\neg p \lor r) & \assum\\
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\begin{subproof}
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\neg p & \assum\\
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\neg p \lor r & $\ori 9$\\
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\bot & $\nege 10, 8$
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\end{subproof}
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p & $\PBC 9-11 $\\
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r & $\impe 12, 7$\\
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\neg p \lor r & $\ori{2} 13$\\
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\bot & $\nege 14, 8$
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\end{subproof}
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\neg p \lor r & $\PBC 8-15$
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\end{logicproof}
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