Question 1208642
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Here is one way to do the derivation. There may be other approaches.
<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>F v ~I</td><td></td><td></td></tr><tr><td>2</td><td>I v H</td><td></td><td></td></tr><tr><td>3</td><td>~(G <--> J) --> ~H</td><td></td><td></td></tr><tr><td>:.</td><td>[ (~G v ~J) & (G v J) ] --> F</td><td></td><td></td></tr><tr><td>4</td><td>~F --> ~I</td><td>1</td><td>Material Implication</td></tr><tr><td>5</td><td>~I --> H</td><td>2</td><td>Material Implication</td></tr><tr><td>6</td><td>~F --> H</td><td>4,5</td><td>Hypothetical Syllogism</td></tr><tr><td>7</td><td>~H --> F</td><td>6</td><td>Transposition</td></tr><tr><td>8</td><td>~(G <--> J) --> F</td><td>3,7</td><td>Hypothetical Syllogism</td></tr><tr><td>9</td><td>[ (~G v ~J) & (G v J) ] --> F</td><td>8</td><td>Material Equivalence</td></tr></table>
Here's a list of <a href="https://www.algebra.com/algebra/homework/Conjunction/logic-rules-of-inference-and-replacement.lesson">rules of inference and replacement</a>
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