Question 1169574
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Here's one way to do the derivation. I'm using arrow symbols in place of horseshoe symbols.


The idea is to assume G is the case and show it leads to (H -> N).
<table border = "1" cellpadding = "5"><tr><td colspan="2">Number</td><td>Statement</td><td>Lines Used</td><td>Reason</td></tr><tr><td>1</td><td></td><td>G -> (E -> N)</td><td></td><td></td></tr><tr><td>2</td><td></td><td>H -> (~N -> E)</td><td></td><td></td></tr><tr><td colspan="2">:.</td><td>G -> (H -> N)</td><td></td><td></td></tr><tr><td></td><td>3</td><td>G</td><td></td><td>Assumption for Conditional Proof</td></tr><tr><td></td><td>4</td><td>E -> N</td><td>1,3</td><td>Modus Ponens</td></tr><tr><td></td><td>5</td><td>(H & ~N) -> E</td><td>2</td><td>Exportation</td></tr><tr><td></td><td>6</td><td>(H & ~N) -> N</td><td>5,4</td><td>Hypothetical Syllogism</td></tr><tr><td></td><td>7</td><td>H -> (~N -> N)</td><td>6</td><td>Exportation</td></tr><tr><td></td><td>8</td><td>H -> (~~N v N)</td><td>7</td><td>Material Implication</td></tr><tr><td></td><td>9</td><td>H -> (N v N)</td><td>8</td><td>Double Negation</td></tr><tr><td></td><td>10</td><td>H -> N</td><td>9</td><td>Tautology</td></tr><tr><td>11</td><td></td><td>G -> (H -> N)</td><td>3-10</td><td>Conditional Proof</td></tr></table>

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