Question 1004972
Note: I'm going to use the symbol <font color=red> -> </font> in place of the horseshoe symbol.



<table border=1><tr><th colspan=2>Number</th><th>Statement</th><th>Lines Used</th><th>Reason</th></tr><tr><td>1</td><td></td><td>(A v B) -&gt; (C v D)</td><td></td><td></td></tr><tr><td>2</td><td></td><td>(C v E) -&gt; D</td><td></td><td></td></tr><tr><td>:.</td><td></td><td>A -&gt; D</td><td></td><td></td></tr><tr><td>|</td><td>3</td><td>A</td><td></td><td>ACP</td></tr><tr><td>|</td><td>4</td><td>A v B</td><td>3</td><td>Add</td></tr><tr><td>|</td><td>5</td><td>C v D</td><td>1,4</td><td>MP</td></tr><tr><td>|</td><td>6</td><td>D v C</td><td>5</td><td>Comm</td></tr><tr><td>|</td><td>7</td><td>~~D v C</td><td>6</td><td>DN</td></tr><tr><td>|</td><td>8</td><td>~D -&gt; C</td><td>7</td><td>MI</td></tr><tr><td>|</td><td>9</td><td>~(C v E) v D</td><td>2</td><td>MI</td></tr><tr><td>|</td><td>10</td><td>(~C * ~E) v D</td><td>9</td><td>DM</td></tr><tr><td>|</td><td>11</td><td>D v (~C * ~E)</td><td>10</td><td>Comm</td></tr><tr><td>|</td><td>12</td><td>(D v ~C) * (D v ~E)</td><td>11</td><td>Dist</td></tr><tr><td>|</td><td>13</td><td>D v ~C</td><td>12</td><td>Simp</td></tr><tr><td>|</td><td>14</td><td>~C v D</td><td>13</td><td>Comm</td></tr><tr><td>|</td><td>15</td><td>C -&gt; D</td><td>14</td><td>MI</td></tr><tr><td>|</td><td>16</td><td>~D -&gt; D</td><td>8,15</td><td>HS</td></tr><tr><td>|</td><td>17</td><td>~~D v D</td><td>16</td><td>MI</td></tr><tr><td>|</td><td>18</td><td>D v D</td><td>17</td><td>DN</td></tr><tr><td>|</td><td>19</td><td>D</td><td>18</td><td>Taut</td></tr><tr><td>20</td><td></td><td>A -&gt; D</td><td>3-19</td><td>CP</td></tr></table>


Abbreviations/Acronyms Used


ACP = Assumption for Conditional Proof
Add = Addition
Comm = Commutation
CP = Conditional Proof
Dist = Distribution
DM = De Morgan's Law
DN = Double Negation
HS = Hypothetical Syllogism
MI = Material Implication
MP = Modus Ponens
Simp = Simplification
Taut = Tautology