Question 1193789
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Here is one way to do the derivation. There are probably more shorter efficient paths to take.<table border = "1" cellpadding = "5"><tr><td>Number</td><td>Statement</td><td>Line(s) Used</td><td>Reason</td></tr><tr><td>1</td><td>H = N</td><td></td><td></td></tr><tr><td>2</td><td>H v N</td><td></td><td></td></tr><tr><td>3</td><td>H -> (N -> U)</td><td></td><td></td></tr><tr><td>:.</td><td>U</td><td></td><td></td></tr><tr><td>4</td><td>(H -> N) & (N -> H)</td><td>1</td><td>Material Equivalence</td></tr><tr><td>5</td><td>H -> N</td><td>4</td><td>Simplification</td></tr><tr><td>6</td><td>N -> H</td><td>4</td><td>Simplification</td></tr><tr><td>7</td><td>~~H v N</td><td>2</td><td>Double Negation</td></tr><tr><td>8</td><td>~H -> N</td><td>7</td><td>Material Implication</td></tr><tr><td>9</td><td>~H -> H</td><td>8, 6</td><td>Hypothetical Syllogism</td></tr><tr><td>10</td><td>~~H v H</td><td>9</td><td>Material Implication</td></tr><tr><td>11</td><td>H v H</td><td>10</td><td>Double Negation</td></tr><tr><td>12</td><td>H</td><td>11</td><td>Tautology</td></tr><tr><td>13</td><td>N v H</td><td>2</td><td>Commutation</td></tr><tr><td>14</td><td>~~N v H</td><td>13</td><td>Double Negation</td></tr><tr><td>15</td><td>~N -> H</td><td>14</td><td>Material Implication</td></tr><tr><td>16</td><td>~N -> N</td><td>15, 5</td><td>Hypothetical Syllogism</td></tr><tr><td>17</td><td>~~N v N</td><td>16</td><td>Material Implication</td></tr><tr><td>18</td><td>N v N</td><td>17</td><td>Double Negation</td></tr><tr><td>19</td><td>N</td><td>18</td><td>Tautology</td></tr><tr><td>20</td><td>N -> U</td><td>3, 12</td><td>Modus Ponens</td></tr><tr><td>21</td><td>U</td><td>20, 19</td><td>Modus Ponens</td></tr></table>I used arrows in place of horseshoe symbols. 
Also, I used a regular equal sign in place of the triple equal sign.
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