document.write( "Question 1009930: Conditional Proof - can use all 18 rules \r
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Algebra.Com's Answer #625431 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
Use a conditional proof twice\r
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Number		Statement	Lines Used	Reason
:.		(P -> Q) <--> (P -> (Q v ~P))
	1	P -> Q		ACP
	2	~P v Q	1	MI
	3	(~P v Q) v ~P	2	Add
	4	~P v (Q v ~P)	3	Assoc
	5	P -> (Q v ~P)	4	MI
6		[P -> Q] -> [P -> (Q v ~P)]	1-5	CP
	7	P -> (Q v ~P)		ACP
	8	~P v (Q v ~P)	7	MI
	9	~P v (~P v Q)	8	Comm
	10	(~P v ~P) v Q	9	Assoc
	11	~P v Q	10	Taut
	12	P -> Q	11	MI
13		[P -> (Q v ~P)] -> [P -> Q]	7-12	CP
14		{[P -> Q] -> [P -> (Q v ~P)]} & {[P -> (Q v ~P)] -> [P -> Q]}	6,13	Conj
15		(P -> Q) <--> (P -> (Q v ~P))	14	ME

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\n" ); document.write( "\n" ); document.write( "ACP = Assumption for Conditional Proof
\n" ); document.write( "Add = Addition
\n" ); document.write( "Assoc = Association
\n" ); document.write( "Comm = Commutation
\n" ); document.write( "Conj = Conjunction
\n" ); document.write( "CP = Conditional Proof
\n" ); document.write( "Dist = Distribution
\n" ); document.write( "DM = De Morgan's Law
\n" ); document.write( "HS = Hypothetical Syllogism
\n" ); document.write( "ME = Material Equivalence
\n" ); document.write( "MI = Material Implication
\n" ); document.write( "MP = Modus Ponens
\n" ); document.write( "Simp = Simplification
\n" ); document.write( "Taut = Tautology \n" ); document.write( "

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