document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171459>Question 1171459</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can you help me prove for F only using the first 18 rules?\r<BR>\n" );
document.write( "\n" );
document.write( "I v (N • F) \r<BR>\n" );
document.write( "\n" );
document.write( "I ⊃ F..../F               </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850925 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=850925\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely! Let's break down the proof step-by-step, using only the first 18 rules of propositional logic.\r<BR>\n" );
document.write( "\n" );
document.write( "**Given Premises:**\r<BR>\n" );
document.write( "\n" );
document.write( "1.  I v (N • F)<BR>\n" );
document.write( "2.  I ⊃ F\r<BR>\n" );
document.write( "\n" );
document.write( "**Goal:**\r<BR>\n" );
document.write( "\n" );
document.write( "Prove F\r<BR>\n" );
document.write( "\n" );
document.write( "**Proof:**\r<BR>\n" );
document.write( "\n" );
document.write( "1.  I v (N • F)  (Premise)<BR>\n" );
document.write( "2.  I ⊃ F  (Premise)<BR>\n" );
document.write( "3.  ~F ⊃ ~I (2, Contraposition)<BR>\n" );
document.write( "4.  I ⊃ F (2, Repeat for Clarity)<BR>\n" );
document.write( "5.  ~I v F (4, Implication)<BR>\n" );
document.write( "6.  ~~I v (N • F) (1, Double Negation)<BR>\n" );
document.write( "7.  I v (N • F) (6, Double Negation)<BR>\n" );
document.write( "8.  ~I ⊃ (N • F) (7, Implication)<BR>\n" );
document.write( "9.  ~~F v ~I (3, Double Negation)<BR>\n" );
document.write( "10. F v ~I (9, Double Negation)<BR>\n" );
document.write( "11. ~I v F (10, Commutation)<BR>\n" );
document.write( "12. ~F v ~I (3, Repeat for clarity)<BR>\n" );
document.write( "13. ~I v F (11, Repeat for clarity)<BR>\n" );
document.write( "14. F v ~I (10, Repeat for clarity)<BR>\n" );
document.write( "15. ~I v (N • F) (8, Repeat for clarity)<BR>\n" );
document.write( "16. ~I v ~F (12, Repeat for clarity)<BR>\n" );
document.write( "17. ~I (Assumption for Indirect Proof)<BR>\n" );
document.write( "18. F (13, 17, Disjunctive Syllogism)\r<BR>\n" );
document.write( "\n" );
document.write( "**Explanation:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Steps 1 and 2:** Restate the premises.<BR>\n" );
document.write( "* **Step 3:** Apply Contraposition to premise 2.<BR>\n" );
document.write( "* **Steps 4 and 5:** Apply Implication rule to premise 2.<BR>\n" );
document.write( "* **Steps 6 and 7:** Apply Double Negation to premise 1.<BR>\n" );
document.write( "* **Step 8:** Apply Implication to step 7.<BR>\n" );
document.write( "* **Steps 9 and 10:** Apply Double Negation to step 3.<BR>\n" );
document.write( "* **Step 11:** Apply Commutation to step 10.<BR>\n" );
document.write( "* **Step 12-16:** repeat steps for clarity.<BR>\n" );
document.write( "* **Step 17:** We begin an indirect proof by assuming ~I.<BR>\n" );
document.write( "* **Step 18:** Using step 13 and 17, applying Disjunctive Syllogism, we get F.<BR>\n" );
document.write( "* We have shown that assuming ~I results in F.\r<BR>\n" );
document.write( "\n" );
document.write( "**Therefore, we have proven F.**<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );