document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171515>Question 1171515</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hey, I hope you are doing well. I have these homework questions that I can't figure out. <BR>\n" );
document.write( "A, ¬F → ¬A ⊢ D → (¬E → F)\r<BR>\n" );
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document.write( "I'm supposed to construct a formal proof\r<BR>\n" );
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document.write( "Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850897 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850897\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Alright, let's break down this logical entailment proof. We need to show that from the premise A, ¬F → ¬A, we can derive the conclusion D → (¬E → F).\r<BR>\n" );
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document.write( "Here's a step-by-step proof using natural deduction:\r<BR>\n" );
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document.write( "**1. A (Premise)**\r<BR>\n" );
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document.write( "**2. ¬F → ¬A (Premise)**\r<BR>\n" );
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document.write( "**3. ¬¬A (Double Negation Introduction, 1)**\r<BR>\n" );
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document.write( "**4. A (Double Negation Elimination, 3)**\r<BR>\n" );
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document.write( "**5. F (Modus Ponens, 2, 4)**\r<BR>\n" );
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document.write( "**6. ¬E → F (Conditional Introduction, 5)** - Here, we discharge the assumption of ¬E.\r<BR>\n" );
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document.write( "**7. D → (¬E → F) (Conditional Introduction, 6)** - Here, we discharge the assumption of D.\r<BR>\n" );
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document.write( "**Explanation:**\r<BR>\n" );
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document.write( "1.  We start with the given premises, A and ¬F → ¬A.<BR>\n" );
document.write( "2.  From A, we can introduce a double negation (¬¬A).<BR>\n" );
document.write( "3.  We can eliminate the double negation to get A.<BR>\n" );
document.write( "4.  Now, we have A, and ¬F → ¬A. Since A is true, ¬A is false. Therefore, ¬F must be false, which means F is true.<BR>\n" );
document.write( "5.  We've derived F.<BR>\n" );
document.write( "6.  To prove ¬E → F, we assume ¬E. Since we've already derived F, the implication ¬E → F is true. We discharge the assumption ¬E.<BR>\n" );
document.write( "7.  To prove D → (¬E → F), we assume D. Since we've already derived ¬E → F, the implication D → (¬E → F) is true. We discharge the assumption D.\r<BR>\n" );
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document.write( "**Therefore, A, ¬F → ¬A ⊢ D → (¬E → F) is a valid entailment.**<BR>\n" );
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