Question 1060204
The idea is to assume the complete opposite of the conclusion (statement 3). 

Then show how that assumption leads to to a contradiction (statement 14). 

This contradiction means that the opposite of the assumption must be true. In other words, the original conclusion is true. 


<table border=1 cellpadding=3><tr><th colspan="2">Number</th><th>Statement</th><th>Lines Used</th><th>Reason</th></tr><tr><td>1</td><td></td><td>B -&gt; (C -&gt; ~B)</td><td></td><td></td></tr><tr><td>2</td><td></td><td>A -&gt; (B -&gt; C)</td><td></td><td></td></tr><tr><td>:.</td><td></td><td>~A v ~B</td><td></td><td></td></tr><tr><td></td><td>3</td><td>~(~A v ~B)</td><td></td><td>AIP</td></tr><tr><td></td><td>4</td><td>~~A &amp; ~~B</td><td>3</td><td>DM</td></tr><tr><td></td><td>5</td><td>A &amp; B</td><td>4</td><td>DN</td></tr><tr><td></td><td>6</td><td>B &amp; A</td><td>5</td><td>Comm</td></tr><tr><td></td><td>7</td><td>A</td><td>5</td><td>Simp</td></tr><tr><td></td><td>8</td><td>B</td><td>6</td><td>Simp</td></tr><tr><td></td><td>9</td><td>C -&gt; ~B</td><td>1,8</td><td>MP</td></tr><tr><td></td><td>10</td><td>B -&gt; C</td><td>2,7</td><td>MP</td></tr><tr><td></td><td>11</td><td>B -&gt; ~B</td><td>10,9</td><td>HS</td></tr><tr><td></td><td>12</td><td>~B v ~B</td><td>11</td><td>MI</td></tr><tr><td></td><td>13</td><td>~B</td><td>12</td><td>Taut</td></tr><tr><td></td><td>14</td><td>B &amp; ~B</td><td>8,13</td><td>Conj</td></tr><tr><td>15</td><td></td><td>~A v ~B</td><td>3-14</td><td>IP</td></tr></table>


Abbreviations/Acronyms Used

AIP = Assumption for Indirect Proof
Comm = Commutation
Conj = Conjunction
DM = De Morgan's Law
DN = Double Negation
HS = Hypothetical Syllogism
IP = Indirect Proof
MI = Material Implication
MP = Modus Ponens
Simp = Simplification
Taut = Tautology