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Theorem List for Intuitionistic Logic Explorer - 1001-1100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsimp1r3 1001 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ (φ ψ χ)) τ η) → χ)

Theoremsimp2l1 1002 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ ((φ ψ χ) θ) η) → φ)

Theoremsimp2l2 1003 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ ((φ ψ χ) θ) η) → ψ)

Theoremsimp2l3 1004 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ ((φ ψ χ) θ) η) → χ)

Theoremsimp2r1 1005 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ (θ (φ ψ χ)) η) → φ)

Theoremsimp2r2 1006 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ (θ (φ ψ χ)) η) → ψ)

Theoremsimp2r3 1007 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ (θ (φ ψ χ)) η) → χ)

Theoremsimp3l1 1008 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ η ((φ ψ χ) θ)) → φ)

Theoremsimp3l2 1009 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ η ((φ ψ χ) θ)) → ψ)

Theoremsimp3l3 1010 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ η ((φ ψ χ) θ)) → χ)

Theoremsimp3r1 1011 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ η (θ (φ ψ χ))) → φ)

Theoremsimp3r2 1012 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ η (θ (φ ψ χ))) → ψ)

Theoremsimp3r3 1013 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ η (θ (φ ψ χ))) → χ)

Theoremsimp11l 1014 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ) χ θ) τ η) → φ)

Theoremsimp11r 1015 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ) χ θ) τ η) → ψ)

Theoremsimp12l 1016 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((χ (φ ψ) θ) τ η) → φ)

Theoremsimp12r 1017 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((χ (φ ψ) θ) τ η) → ψ)

Theoremsimp13l 1018 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((χ θ (φ ψ)) τ η) → φ)

Theoremsimp13r 1019 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((χ θ (φ ψ)) τ η) → ψ)

Theoremsimp21l 1020 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ ((φ ψ) χ θ) η) → φ)

Theoremsimp21r 1021 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ ((φ ψ) χ θ) η) → ψ)

Theoremsimp22l 1022 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ (χ (φ ψ) θ) η) → φ)

Theoremsimp22r 1023 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ (χ (φ ψ) θ) η) → ψ)

Theoremsimp23l 1024 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ (χ θ (φ ψ)) η) → φ)

Theoremsimp23r 1025 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ (χ θ (φ ψ)) η) → ψ)

Theoremsimp31l 1026 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ η ((φ ψ) χ θ)) → φ)

Theoremsimp31r 1027 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ η ((φ ψ) χ θ)) → ψ)

Theoremsimp32l 1028 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ η (χ (φ ψ) θ)) → φ)

Theoremsimp32r 1029 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ η (χ (φ ψ) θ)) → ψ)

Theoremsimp33l 1030 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ η (χ θ (φ ψ))) → φ)

Theoremsimp33r 1031 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((τ η (χ θ (φ ψ))) → ψ)

Theoremsimp111 1032 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ χ) θ τ) η ζ) → φ)

Theoremsimp112 1033 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ χ) θ τ) η ζ) → ψ)

Theoremsimp113 1034 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((φ ψ χ) θ τ) η ζ) → χ)

Theoremsimp121 1035 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ (φ ψ χ) τ) η ζ) → φ)

Theoremsimp122 1036 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ (φ ψ χ) τ) η ζ) → ψ)

Theoremsimp123 1037 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ (φ ψ χ) τ) η ζ) → χ)

Theoremsimp131 1038 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ τ (φ ψ χ)) η ζ) → φ)

Theoremsimp132 1039 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ τ (φ ψ χ)) η ζ) → ψ)

Theoremsimp133 1040 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((θ τ (φ ψ χ)) η ζ) → χ)

Theoremsimp211 1041 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ((φ ψ χ) θ τ) ζ) → φ)

Theoremsimp212 1042 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ((φ ψ χ) θ τ) ζ) → ψ)

Theoremsimp213 1043 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ((φ ψ χ) θ τ) ζ) → χ)

Theoremsimp221 1044 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η (θ (φ ψ χ) τ) ζ) → φ)

Theoremsimp222 1045 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η (θ (φ ψ χ) τ) ζ) → ψ)

Theoremsimp223 1046 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η (θ (φ ψ χ) τ) ζ) → χ)

Theoremsimp231 1047 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η (θ τ (φ ψ χ)) ζ) → φ)

Theoremsimp232 1048 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η (θ τ (φ ψ χ)) ζ) → ψ)

Theoremsimp233 1049 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η (θ τ (φ ψ χ)) ζ) → χ)

Theoremsimp311 1050 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ ((φ ψ χ) θ τ)) → φ)

Theoremsimp312 1051 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ ((φ ψ χ) θ τ)) → ψ)

Theoremsimp313 1052 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ ((φ ψ χ) θ τ)) → χ)

Theoremsimp321 1053 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ (θ (φ ψ χ) τ)) → φ)

Theoremsimp322 1054 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ (θ (φ ψ χ) τ)) → ψ)

Theoremsimp323 1055 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ (θ (φ ψ χ) τ)) → χ)

Theoremsimp331 1056 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ (θ τ (φ ψ χ))) → φ)

Theoremsimp332 1057 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ (θ τ (φ ψ χ))) → ψ)

Theoremsimp333 1058 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ (θ τ (φ ψ χ))) → χ)

Theorem3adantl1 1059 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((φ ψ) χ) → θ)       (((τ φ ψ) χ) → θ)

Theorem3adantl2 1060 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((φ ψ) χ) → θ)       (((φ τ ψ) χ) → θ)

Theorem3adantl3 1061 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((φ ψ) χ) → θ)       (((φ ψ τ) χ) → θ)

Theorem3adantr1 1062 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((φ (ψ χ)) → θ)       ((φ (τ ψ χ)) → θ)

Theorem3adantr2 1063 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((φ (ψ χ)) → θ)       ((φ (ψ τ χ)) → θ)

Theorem3adantr3 1064 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((φ (ψ χ)) → θ)       ((φ (ψ χ τ)) → θ)

((φ χ) → θ)       (((φ ψ τ) χ) → θ)

((φ χ) → θ)       (((ψ φ τ) χ) → θ)

((φ χ) → θ)       (((ψ τ φ) χ) → θ)

Theorem3ad2antr1 1068 Deduction adding a conjuncts to antecedent. (Contributed by NM, 25-Dec-2007.)
((φ χ) → θ)       ((φ (χ ψ τ)) → θ)

Theorem3ad2antr2 1069 Deduction adding a conjuncts to antecedent. (Contributed by NM, 27-Dec-2007.)
((φ χ) → θ)       ((φ (ψ χ τ)) → θ)

Theorem3ad2antr3 1070 Deduction adding a conjuncts to antecedent. (Contributed by NM, 30-Dec-2007.)
((φ χ) → θ)       ((φ (ψ τ χ)) → θ)

Theorem3anibar 1071 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 1072 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(φ → (φ ψ χ))

Theorem3mix2 1073 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(φ → (ψ φ χ))

Theorem3mix3 1074 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(φ → (ψ χ φ))

Theorem3mix1i 1075 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
φ       (φ ψ χ)

Theorem3mix2i 1076 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
φ       (ψ φ χ)

Theorem3mix3i 1077 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
φ       (ψ χ φ)

Theorem3mix1d 1078 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
(φψ)       (φ → (ψ χ θ))

Theorem3mix2d 1079 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
(φψ)       (φ → (χ ψ θ))

Theorem3mix3d 1080 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
(φψ)       (φ → (χ θ ψ))

Theorem3pm3.2i 1081 Infer conjunction of premises. (Contributed by NM, 10-Feb-1995.)
φ    &   ψ    &   χ       (φ ψ χ)

Theorempm3.2an3 1082 pm3.2 126 for a triple conjunction. (Contributed by Alan Sare, 24-Oct-2011.)
(φ → (ψ → (χ → (φ ψ χ))))

Theorem3jca 1083 Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.)
(φψ)    &   (φχ)    &   (φθ)       (φ → (ψ χ θ))

Theorem3jcad 1084 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 1085 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 1086 Detach a conjunction of truths in a biconditional. (Contributed by Mario Carneiro, 11-May-2014.)
(φχ)    &   (φθ)    &   (φτ)    &   (φ → (ψ ↔ (χ θ τ)))       (φψ)

Theoremsyl3anbrc 1087 Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014.)
(φψ)    &   (φχ)    &   (φθ)    &   (τ ↔ (ψ χ θ))       (φτ)

Theorem3anim123i 1088 Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.)
(φψ)    &   (χθ)    &   (τη)       ((φ χ τ) → (ψ θ η))

Theorem3anim1i 1089 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 16-Aug-2009.)
(φψ)       ((φ χ θ) → (ψ χ θ))

Theorem3anim2i 1090 Add two conjuncts to antecedent and consequent. (Contributed by AV, 21-Nov-2019.)
(φψ)       ((χ φ θ) → (χ ψ θ))

Theorem3anim3i 1091 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 19-Aug-2009.)
(φψ)       ((χ θ φ) → (χ θ ψ))

Theorem3anbi123i 1092 Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994.)
(φψ)    &   (χθ)    &   (τη)       ((φ χ τ) ↔ (ψ θ η))

Theorem3orbi123i 1093 Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)
(φψ)    &   (χθ)    &   (τη)       ((φ χ τ) ↔ (ψ θ η))

Theorem3anbi1i 1094 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(φψ)       ((φ χ θ) ↔ (ψ χ θ))

Theorem3anbi2i 1095 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(φψ)       ((χ φ θ) ↔ (χ ψ θ))

Theorem3anbi3i 1096 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(φψ)       ((χ θ φ) ↔ (χ θ ψ))

Theorem3imp 1097 Importation inference. (Contributed by NM, 8-Apr-1994.)
(φ → (ψ → (χθ)))       ((φ ψ χ) → θ)

Theorem3impa 1098 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
(((φ ψ) χ) → θ)       ((φ ψ χ) → θ)

Theorem3impb 1099 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
((φ (ψ χ)) → θ)       ((φ ψ χ) → θ)

Theorem3impia 1100 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
((φ ψ) → (χθ))       ((φ ψ χ) → θ)

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