SOLUTION: Conditional Proof - can use all 18 rules (P -> Q) <-> (P -> (Q v ~P))

Question 1009930: Conditional Proof - can use all 18 rules
(P -> Q) <-> (P -> (Q v ~P))
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Use a conditional proof twice

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

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