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Theorem List for Intuitionistic Logic Explorer - 1001-1100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsimp11l 1001 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ) χ θ) τ η) → φ)
 
Theoremsimp11r 1002 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ) χ θ) τ η) → ψ)
 
Theoremsimp12l 1003 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((χ (φ ψ) θ) τ η) → φ)
 
Theoremsimp12r 1004 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((χ (φ ψ) θ) τ η) → ψ)
 
Theoremsimp13l 1005 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((χ θ (φ ψ)) τ η) → φ)
 
Theoremsimp13r 1006 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((χ θ (φ ψ)) τ η) → ψ)
 
Theoremsimp21l 1007 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ ((φ ψ) χ θ) η) → φ)
 
Theoremsimp21r 1008 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ ((φ ψ) χ θ) η) → ψ)
 
Theoremsimp22l 1009 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ (χ (φ ψ) θ) η) → φ)
 
Theoremsimp22r 1010 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ (χ (φ ψ) θ) η) → ψ)
 
Theoremsimp23l 1011 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ (χ θ (φ ψ)) η) → φ)
 
Theoremsimp23r 1012 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ (χ θ (φ ψ)) η) → ψ)
 
Theoremsimp31l 1013 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ η ((φ ψ) χ θ)) → φ)
 
Theoremsimp31r 1014 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ η ((φ ψ) χ θ)) → ψ)
 
Theoremsimp32l 1015 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ η (χ (φ ψ) θ)) → φ)
 
Theoremsimp32r 1016 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ η (χ (φ ψ) θ)) → ψ)
 
Theoremsimp33l 1017 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ η (χ θ (φ ψ))) → φ)
 
Theoremsimp33r 1018 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ η (χ θ (φ ψ))) → ψ)
 
Theoremsimp111 1019 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ χ) θ τ) η ζ) → φ)
 
Theoremsimp112 1020 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ χ) θ τ) η ζ) → ψ)
 
Theoremsimp113 1021 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ χ) θ τ) η ζ) → χ)
 
Theoremsimp121 1022 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ (φ ψ χ) τ) η ζ) → φ)
 
Theoremsimp122 1023 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ (φ ψ χ) τ) η ζ) → ψ)
 
Theoremsimp123 1024 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ (φ ψ χ) τ) η ζ) → χ)
 
Theoremsimp131 1025 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ τ (φ ψ χ)) η ζ) → φ)
 
Theoremsimp132 1026 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ τ (φ ψ χ)) η ζ) → ψ)
 
Theoremsimp133 1027 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ τ (φ ψ χ)) η ζ) → χ)
 
Theoremsimp211 1028 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ((φ ψ χ) θ τ) ζ) → φ)
 
Theoremsimp212 1029 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ((φ ψ χ) θ τ) ζ) → ψ)
 
Theoremsimp213 1030 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ((φ ψ χ) θ τ) ζ) → χ)
 
Theoremsimp221 1031 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η (θ (φ ψ χ) τ) ζ) → φ)
 
Theoremsimp222 1032 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η (θ (φ ψ χ) τ) ζ) → ψ)
 
Theoremsimp223 1033 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η (θ (φ ψ χ) τ) ζ) → χ)
 
Theoremsimp231 1034 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η (θ τ (φ ψ χ)) ζ) → φ)
 
Theoremsimp232 1035 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η (θ τ (φ ψ χ)) ζ) → ψ)
 
Theoremsimp233 1036 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η (θ τ (φ ψ χ)) ζ) → χ)
 
Theoremsimp311 1037 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ ((φ ψ χ) θ τ)) → φ)
 
Theoremsimp312 1038 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ ((φ ψ χ) θ τ)) → ψ)
 
Theoremsimp313 1039 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ ((φ ψ χ) θ τ)) → χ)
 
Theoremsimp321 1040 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ (θ (φ ψ χ) τ)) → φ)
 
Theoremsimp322 1041 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ (θ (φ ψ χ) τ)) → ψ)
 
Theoremsimp323 1042 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ (θ (φ ψ χ) τ)) → χ)
 
Theoremsimp331 1043 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ (θ τ (φ ψ χ))) → φ)
 
Theoremsimp332 1044 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ (θ τ (φ ψ χ))) → ψ)
 
Theoremsimp333 1045 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ (θ τ (φ ψ χ))) → χ)
 
Theorem3adantl1 1046 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((φ ψ) χ) → θ)       (((τ φ ψ) χ) → θ)
 
Theorem3adantl2 1047 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((φ ψ) χ) → θ)       (((φ τ ψ) χ) → θ)
 
Theorem3adantl3 1048 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((φ ψ) χ) → θ)       (((φ ψ τ) χ) → θ)
 
Theorem3adantr1 1049 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((φ (ψ χ)) → θ)       ((φ (τ ψ χ)) → θ)
 
Theorem3adantr2 1050 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((φ (ψ χ)) → θ)       ((φ (ψ τ χ)) → θ)
 
Theorem3adantr3 1051 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((φ (ψ χ)) → θ)       ((φ (ψ χ τ)) → θ)
 
Theorem3ad2antl1 1052 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((φ χ) → θ)       (((φ ψ τ) χ) → θ)
 
Theorem3ad2antl2 1053 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((φ χ) → θ)       (((ψ φ τ) χ) → θ)
 
Theorem3ad2antl3 1054 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((φ χ) → θ)       (((ψ τ φ) χ) → θ)
 
Theorem3ad2antr1 1055 Deduction adding a conjuncts to antecedent. (Contributed by NM, 25-Dec-2007.)
((φ χ) → θ)       ((φ (χ ψ τ)) → θ)
 
Theorem3ad2antr2 1056 Deduction adding a conjuncts to antecedent. (Contributed by NM, 27-Dec-2007.)
((φ χ) → θ)       ((φ (ψ χ τ)) → θ)
 
Theorem3ad2antr3 1057 Deduction adding a conjuncts to antecedent. (Contributed by NM, 30-Dec-2007.)
((φ χ) → θ)       ((φ (ψ τ χ)) → θ)
 
Theorem3anibar 1058 Remove a hypothesis from the second member of a biimplication. (Contributed by FL, 22-Jul-2008.)
((φ ψ χ) → (θ ↔ (χ τ)))       ((φ ψ χ) → (θτ))
 
Theorem3mix1 1059 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(φ → (φ ψ χ))
 
Theorem3mix2 1060 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(φ → (ψ φ χ))
 
Theorem3mix3 1061 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(φ → (ψ χ φ))
 
Theorem3mix1i 1062 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
φ       (φ ψ χ)
 
Theorem3mix2i 1063 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
φ       (ψ φ χ)
 
Theorem3mix3i 1064 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
φ       (ψ χ φ)
 
Theorem3pm3.2i 1065 Infer conjunction of premises. (Contributed by NM, 10-Feb-1995.)
φ    &   ψ    &   χ       (φ ψ χ)
 
Theorempm3.2an3 1066 pm3.2 126 for a triple conjunction. (Contributed by Alan Sare, 24-Oct-2011.)
(φ → (ψ → (χ → (φ ψ χ))))
 
Theorem3jca 1067 Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.)
(φψ)    &   (φχ)    &   (φθ)       (φ → (ψ χ θ))
 
Theorem3jcad 1068 Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005.)
(φ → (ψχ))    &   (φ → (ψθ))    &   (φ → (ψτ))       (φ → (ψ → (χ θ τ)))
 
Theoremmpbir3an 1069 Detach a conjunction of truths in a biconditional. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 9-Jan-2015.)
ψ    &   χ    &   θ    &   (φ ↔ (ψ χ θ))       φ
 
Theoremmpbir3and 1070 Detach a conjunction of truths in a biconditional. (Contributed by Mario Carneiro, 11-May-2014.)
(φχ)    &   (φθ)    &   (φτ)    &   (φ → (ψ ↔ (χ θ τ)))       (φψ)
 
Theoremmpbir3anOLD 1071 Obsolete version of mpbir3an 1069 as of 9-Jan-2015. (Contributed by NM, 16-Sep-2011.)
(φ ↔ (ψ χ θ))    &   ψ    &   χ    &   θ       φ
 
Theoremmpbir3andOLD 1072 Obsolete version of mpbir3and 1070 as of 9-Jan-2015. (Contributed by NM, 11-May-2014.)
(φ → (ψ ↔ (χ θ τ)))    &   (φχ)    &   (φθ)    &   (φτ)       (φψ)
 
Theoremsyl3anbrc 1073 Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014.)
(φψ)    &   (φχ)    &   (φθ)    &   (τ ↔ (ψ χ θ))       (φτ)
 
Theorem3anim123i 1074 Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.)
(φψ)    &   (χθ)    &   (τη)       ((φ χ τ) → (ψ θ η))
 
Theorem3anim1i 1075 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 16-Aug-2009.)
(φψ)       ((φ χ θ) → (ψ χ θ))
 
Theorem3anim3i 1076 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 19-Aug-2009.)
(φψ)       ((χ θ φ) → (χ θ ψ))
 
Theorem3anbi123i 1077 Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994.)
(φψ)    &   (χθ)    &   (τη)       ((φ χ τ) ↔ (ψ θ η))
 
Theorem3orbi123i 1078 Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)
(φψ)    &   (χθ)    &   (τη)       ((φ χ τ) ↔ (ψ θ η))
 
Theorem3anbi1i 1079 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(φψ)       ((φ χ θ) ↔ (ψ χ θ))
 
Theorem3anbi2i 1080 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(φψ)       ((χ φ θ) ↔ (χ ψ θ))
 
Theorem3anbi3i 1081 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(φψ)       ((χ θ φ) ↔ (χ θ ψ))
 
Theorem3imp 1082 Importation inference. (Contributed by NM, 8-Apr-1994.)
(φ → (ψ → (χθ)))       ((φ ψ χ) → θ)
 
Theorem3impa 1083 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
(((φ ψ) χ) → θ)       ((φ ψ χ) → θ)
 
Theorem3impb 1084 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
((φ (ψ χ)) → θ)       ((φ ψ χ) → θ)
 
Theorem3impia 1085 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
((φ ψ) → (χθ))       ((φ ψ χ) → θ)
 
Theorem3impib 1086 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
(φ → ((ψ χ) → θ))       ((φ ψ χ) → θ)
 
Theorem3exp 1087 Exportation inference. (Contributed by NM, 30-May-1994.)
((φ ψ χ) → θ)       (φ → (ψ → (χθ)))
 
Theorem3expa 1088 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
((φ ψ χ) → θ)       (((φ ψ) χ) → θ)
 
Theorem3expb 1089 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
((φ ψ χ) → θ)       ((φ (ψ χ)) → θ)
 
Theorem3expia 1090 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
((φ ψ χ) → θ)       ((φ ψ) → (χθ))
 
Theorem3expib 1091 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
((φ ψ χ) → θ)       (φ → ((ψ χ) → θ))
 
Theorem3com12 1092 Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM, 28-Jan-1996.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((φ ψ χ) → θ)       ((ψ φ χ) → θ)
 
Theorem3com13 1093 Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM, 28-Jan-1996.)
((φ ψ χ) → θ)       ((χ ψ φ) → θ)
 
Theorem3com23 1094 Commutation in antecedent. Swap 2nd and 3rd. (Contributed by NM, 28-Jan-1996.)
((φ ψ χ) → θ)       ((φ χ ψ) → θ)
 
Theorem3coml 1095 Commutation in antecedent. Rotate left. (Contributed by NM, 28-Jan-1996.)
((φ ψ χ) → θ)       ((ψ χ φ) → θ)
 
Theorem3comr 1096 Commutation in antecedent. Rotate right. (Contributed by NM, 28-Jan-1996.)
((φ ψ χ) → θ)       ((χ φ ψ) → θ)
 
Theorem3adant3r1 1097 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Feb-2008.)
((φ ψ χ) → θ)       ((φ (τ ψ χ)) → θ)
 
Theorem3adant3r2 1098 Deduction adding a conjunct to antecedent. (Contributed by NM, 17-Feb-2008.)
((φ ψ χ) → θ)       ((φ (ψ τ χ)) → θ)
 
Theorem3adant3r3 1099 Deduction adding a conjunct to antecedent. (Contributed by NM, 18-Feb-2008.)
((φ ψ χ) → θ)       ((φ (ψ χ τ)) → θ)
 
Theorem3an1rs 1100 Swap conjuncts. (Contributed by NM, 16-Dec-2007.)
(((φ ψ χ) θ) → τ)       (((φ ψ θ) χ) → τ)
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