Premises
(comma separated)
Conclusion
⊦
Enter your proof below then
or
(check for primitive rule help only)
1 (1) PvQ->R A 2 (2) P A 2 (3) PvQ 2 vI 1,2 (4) R 1,3 ->E 5 (5) @xFx A 5 (6) Fa 5 @E 1,2,5 (7) Fa&R 4,6 &I 8 (8) Fa&R->S A 1,2,8,5 (9) S 7,8 ->E
Rule : Annotation : Pattern
[P] represents assume P
Replace appropriate line numbers
--------------------- Primitive Rules ---------------------
A (Assumption) : A : any wff
&I (Ampersand Intro): 1,2 &I : P, Q |- P & Q
&E (Ampersand Elim): 1 &E : P & Q |- P
&E (Ampersand Elim): 1 &E : P & Q |- Q
vI (Wedge Intro): 1 vI : P |- P v Q
vE (Wedge Elim) : 1,2 vE : P v Q, ~P |- Q
vE (Wedge Elim) : 1,2 vE : P v Q, ~Q |- P
->I (Arrow Intro): 2 ->I(1) : [P]...Q |- P -> Q
->E (Arrow Elim): 1,2 ->E : P -> Q, P |- Q
RAA (Reductio ad Absurdum): 2,3 RAA(1): [P] ...Q...~Q |- ~P
RAA (Reductio ad Absurdum): 2,3 RAA(1): [~P] ...Q...~Q |- P
RAA (Reductio ad Absurdum): 2,3 RAA(1): [P] ...~P |- ~P
RAA (Reductio ad Absurdum): 2,3 RAA(1): [~P] ...P |- P
<->I (Double-Arrow Intro): 1,2 <->I : P -> Q, Q -> P |- P <-> Q
<->E (Double-Arrow Elim): 1 <->E : P <-> Q |- P -> Q
<->E (Double-Arrow Elim): 1 <->E : P <-> Q |- Q -> P
@E (Universal Elim): 1 @E : @xFx |- Fa
@I (Universal Intro): 1 @I : Fa |- @xFx
$I (Existential Intro): 1 $I : Fa |- $xFx
$E (Existential Elim): 1,3 $E(2) : $xFx, [Fa]...P |- P
=I (Identity Intro): =I : |- a = a
=E (Identity Elim): 1,2 =E : Fa, a = b |- Fb
--------------------- Derived Rules ---------------------
DN (Double Negation) : 1 DN : ~~P -||- P
MTT (Modus Tollendo Tollens): 1,2 MTT : P -> Q, ~Q |- ~P
HS (Hypothetical Syllogism) : 1,2 HS : P -> Q, Q -> R |- P -> R
TC (True Consequent) : 1 TC : P |- Q -> P
FA (False Antecedent): 1 FA : ~P |- P -> Q
FA (False Antecedent): 1 FA : P |- ~P -> Q
IA (Impossible Antecedent) : 1, 2 IA : P -> Q, P -> ~Q |- ~P
v->(Wedge Arrow) : 1 v-> : ~P v Q -||- P -> Q
v->(Wedge Arrow) : 1 v-> : P v Q -||- ~P -> Q
v->(Wedge Arrow) : 1 v-> : P v Q -||- ~Q -> P
v->(Wedge Arrow) : 1 v-> : P v ~Q -||- Q -> P
SimDil (Simple Dilemma) : 1,2,3 SimDil : P v Q, P -> R, Q -> R |- R
ComDil (Complex Dilemma): 1,2,3 ComDil : P v Q, P -> R, Q -> S |- R v S
SpecDil (Special Dilemma): 1,2 SpecDil : P -> Q, ~P -> Q |- Q
DM (DeMorgan) : 1 DM : ~(P v Q) -||- ~P & ~Q
DM (DeMorgan) : 1 DM : ~(P & Q) -||- ~P v ~Q
DM (DeMorgan) : 1 DM : P v Q -||- ~(~P & ~Q)
DM (DeMorgan) : 1 DM : P & Q -||- ~(~P v ~Q)
Neg-> (Negated Arrow) : 1 Neg-> : ~(P->Q) -||- P&~Q
Neg-> (Negated Arrow) : 1 Neg-> : ~(P->~Q) -||- P&Q
Neg-> (Negated Arrow) : 1 Neg-> : P->Q -||- ~(P&~Q)
Neg-> (Negated Arrow) : 1 Neg-> : P->~Q -||- ~(P&Q)
&Comm (& Commutativity) : 1 &Comm : P & Q -||- Q & P
vComm (v Commutativity) : 1 vComm : P v Q -||- Q v P
<->Comm (<->Commutativity): P <-> Q -||- Q <-> P
Trans : (Transposition) : 1 Trans : P -> Q -||- ~Q -> ~P
&Assoc (& Associativity) : 1 &Assoc : P & (Q & R) -||- (P & Q) & R
vAssoc (v Associativity) : 1 vAssoc : P v (Q v R) -||- (Q v P) v R
&/vDist (&/v Distribution) : 1 &/v Dist : P & (Q v R) -||- (P & Q) v (P & R)
v/&Dist (v/& Distribution) : 1 v/& Dist : P v (Q & R) -||- (P v Q) & (P v R)
Imp/Exp (Imp/Exportation) 1 Imp/Exp : P -> (Q -> R) -||- P & Q -> R
BP (Biconditional Ponens) : 1,2 BP : P <-> Q, P |- Q
BP (Biconditional Ponens) : 1,2 BP : P <-> Q, Q |- P
BT (Biconditional Tollens) : 1,2 BP : P <-> Q, ~P |- ~Q
BT (Biconditional Tollens) : 1,2 BT : P <-> Q, ~Q |- ~P
BiTrans (BiTransposition) : 1 BiTrans : P <-> Q -||- ~Q <-> ~P
BiTrans (BiTransposition) : 1 BiTrans : P <-> ~Q -||- ~P <-> Q
Neg<-> (Negated<->) : 1 Neg<-> : ~(P <-> Q) -||- P <-> ~Q
Neg<-> (Negated<->) : 1 Neg<-> : ~(P <-> Q) -||- ~P <-> Q
Quantifier Exchange (QE) : 1 QE : ~@xFx -||- $x~Fx
Quantifier Exchange (QE) : 1 QE : ~$xFx -||- @x~Fx
Quantifier Exchange (QE) : 1 QE : ~@~xFx -||- $xFx
Quantifier Exchange (QE) : 1 QE : ~$x~Fx -||- @xFx
You can apply primitive rules in a short form using "do" statements
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