Question 1031401
<table border=1 cellpadding=3><tr><th>Number</th><th>Statement</th><th>Lines Used</th><th>Reason</th></tr><tr><td>1</td><td>~A &amp; ~B</td><td></td><td></td></tr><tr><td>2</td><td>~D -&gt; A</td><td></td><td></td></tr><tr><td>3</td><td>M -&gt; [(N v O) -&gt; P]</td><td></td><td></td></tr><tr><td>4</td><td>Q -&gt; (S v T)</td><td></td><td></td></tr><tr><td>5</td><td>(~Q v ~R) -&gt; (M &amp; N)</td><td></td><td></td></tr><tr><td>6</td><td>~D v ~(S v T)</td><td></td><td></td></tr><tr><td>:.</td><td>P &amp; ~B</td><td></td><td></td></tr><tr><td>7</td><td>~B &amp; ~A</td><td>1</td><td>Commutation</td></tr><tr><td>8</td><td>~A</td><td>1</td><td>Simplification</td></tr><tr><td>9</td><td>~B</td><td>7</td><td>Simplification</td></tr><tr><td>10</td><td>~~D</td><td>2,8</td><td>Modus Tollens</td></tr><tr><td>11</td><td>~(S v T)</td><td>6,10</td><td>Disjunctive Syllogism</td></tr><tr><td>12</td><td>~Q</td><td>4,11</td><td>Modus Tollens</td></tr><tr><td>13</td><td>~Q v ~R</td><td>12</td><td>Addition</td></tr><tr><td>14</td><td>M &amp; N</td><td>5,13</td><td>Modus Ponens</td></tr><tr><td>15</td><td>N &amp; M</td><td>14</td><td>Commutation</td></tr><tr><td>16</td><td>M</td><td>14</td><td>Simplification</td></tr><tr><td>17</td><td>N</td><td>15</td><td>Simplification</td></tr><tr><td>18</td><td>N v O</td><td>17</td><td>Addition</td></tr><tr><td>19</td><td>(N v O) -&gt; P</td><td>3,16</td><td>Modus Ponens</td></tr><tr><td>20</td><td>P</td><td>19,18</td><td>Modus Ponens</td></tr><tr><td>21</td><td>P &amp; ~B</td><td>20,9</td><td>Conjunction</td></tr></table>