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Theorem List for Intuitionistic Logic Explorer - 1101-1200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem3impib 1101 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
(φ → ((ψ χ) → θ))       ((φ ψ χ) → θ)

Theorem3exp 1102 Exportation inference. (Contributed by NM, 30-May-1994.)
((φ ψ χ) → θ)       (φ → (ψ → (χθ)))

Theorem3expa 1103 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
((φ ψ χ) → θ)       (((φ ψ) χ) → θ)

Theorem3expb 1104 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
((φ ψ χ) → θ)       ((φ (ψ χ)) → θ)

Theorem3expia 1105 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
((φ ψ χ) → θ)       ((φ ψ) → (χθ))

Theorem3expib 1106 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
((φ ψ χ) → θ)       (φ → ((ψ χ) → θ))

Theorem3com12 1107 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 1108 Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM, 28-Jan-1996.)
((φ ψ χ) → θ)       ((χ ψ φ) → θ)

Theorem3com23 1109 Commutation in antecedent. Swap 2nd and 3rd. (Contributed by NM, 28-Jan-1996.)
((φ ψ χ) → θ)       ((φ χ ψ) → θ)

Theorem3coml 1110 Commutation in antecedent. Rotate left. (Contributed by NM, 28-Jan-1996.)
((φ ψ χ) → θ)       ((ψ χ φ) → θ)

Theorem3comr 1111 Commutation in antecedent. Rotate right. (Contributed by NM, 28-Jan-1996.)
((φ ψ χ) → θ)       ((χ φ ψ) → θ)

Theorem3adant3r1 1112 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Feb-2008.)
((φ ψ χ) → θ)       ((φ (τ ψ χ)) → θ)

Theorem3adant3r2 1113 Deduction adding a conjunct to antecedent. (Contributed by NM, 17-Feb-2008.)
((φ ψ χ) → θ)       ((φ (ψ τ χ)) → θ)

Theorem3adant3r3 1114 Deduction adding a conjunct to antecedent. (Contributed by NM, 18-Feb-2008.)
((φ ψ χ) → θ)       ((φ (ψ χ τ)) → θ)

Theorem3an1rs 1115 Swap conjuncts. (Contributed by NM, 16-Dec-2007.)
(((φ ψ χ) θ) → τ)       (((φ ψ θ) χ) → τ)

Theorem3imp1 1116 Importation to left triple conjunction. (Contributed by NM, 24-Feb-2005.)
(φ → (ψ → (χ → (θτ))))       (((φ ψ χ) θ) → τ)

Theorem3impd 1117 Importation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.)
(φ → (ψ → (χ → (θτ))))       (φ → ((ψ χ θ) → τ))

Theorem3imp2 1118 Importation to right triple conjunction. (Contributed by NM, 26-Oct-2006.)
(φ → (ψ → (χ → (θτ))))       ((φ (ψ χ θ)) → τ)

Theorem3exp1 1119 Exportation from left triple conjunction. (Contributed by NM, 24-Feb-2005.)
(((φ ψ χ) θ) → τ)       (φ → (ψ → (χ → (θτ))))

Theorem3expd 1120 Exportation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.)
(φ → ((ψ χ θ) → τ))       (φ → (ψ → (χ → (θτ))))

Theorem3exp2 1121 Exportation from right triple conjunction. (Contributed by NM, 26-Oct-2006.)
((φ (ψ χ θ)) → τ)       (φ → (ψ → (χ → (θτ))))

Theoremexp5o 1122 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
((φ ψ χ) → ((θ τ) → η))       (φ → (ψ → (χ → (θ → (τη)))))

Theoremexp516 1123 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(((φ (ψ χ θ)) τ) → η)       (φ → (ψ → (χ → (θ → (τη)))))

Theoremexp520 1124 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(((φ ψ χ) (θ τ)) → η)       (φ → (ψ → (χ → (θ → (τη)))))

Theorem3anassrs 1125 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by Mario Carneiro, 4-Jan-2017.)
((φ (ψ χ θ)) → τ)       ((((φ ψ) χ) θ) → τ)

Theorem3adant1l 1126 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((φ ψ χ) → θ)       (((τ φ) ψ χ) → θ)

Theorem3adant1r 1127 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((φ ψ χ) → θ)       (((φ τ) ψ χ) → θ)

Theorem3adant2l 1128 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((φ ψ χ) → θ)       ((φ (τ ψ) χ) → θ)

Theorem3adant2r 1129 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((φ ψ χ) → θ)       ((φ (ψ τ) χ) → θ)

Theorem3adant3l 1130 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((φ ψ χ) → θ)       ((φ ψ (τ χ)) → θ)

Theorem3adant3r 1131 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((φ ψ χ) → θ)       ((φ ψ (χ τ)) → θ)

Theoremsyl12anc 1132 Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
(φψ)    &   (φχ)    &   (φθ)    &   ((ψ (χ θ)) → τ)       (φτ)

Theoremsyl21anc 1133 Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
(φψ)    &   (φχ)    &   (φθ)    &   (((ψ χ) θ) → τ)       (φτ)

Theoremsyl3anc 1134 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   ((ψ χ θ) → τ)       (φτ)

Theoremsyl22anc 1135 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (((ψ χ) (θ τ)) → η)       (φη)

Theoremsyl13anc 1136 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   ((ψ (χ θ τ)) → η)       (φη)

Theoremsyl31anc 1137 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (((ψ χ θ) τ) → η)       (φη)

Theoremsyl112anc 1138 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   ((ψ χ (θ τ)) → η)       (φη)

Theoremsyl121anc 1139 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   ((ψ (χ θ) τ) → η)       (φη)

Theoremsyl211anc 1140 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (((ψ χ) θ τ) → η)       (φη)

Theoremsyl23anc 1141 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (((ψ χ) (θ τ η)) → ζ)       (φζ)

Theoremsyl32anc 1142 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (((ψ χ θ) (τ η)) → ζ)       (φζ)

Theoremsyl122anc 1143 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   ((ψ (χ θ) (τ η)) → ζ)       (φζ)

Theoremsyl212anc 1144 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (((ψ χ) θ (τ η)) → ζ)       (φζ)

Theoremsyl221anc 1145 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (((ψ χ) (θ τ) η) → ζ)       (φζ)

Theoremsyl113anc 1146 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   ((ψ χ (θ τ η)) → ζ)       (φζ)

Theoremsyl131anc 1147 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   ((ψ (χ θ τ) η) → ζ)       (φζ)

Theoremsyl311anc 1148 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (((ψ χ θ) τ η) → ζ)       (φζ)

Theoremsyl33anc 1149 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (φζ)    &   (((ψ χ θ) (τ η ζ)) → σ)       (φσ)

Theoremsyl222anc 1150 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (φζ)    &   (((ψ χ) (θ τ) (η ζ)) → σ)       (φσ)

Theoremsyl123anc 1151 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (φζ)    &   ((ψ (χ θ) (τ η ζ)) → σ)       (φσ)

Theoremsyl132anc 1152 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (φζ)    &   ((ψ (χ θ τ) (η ζ)) → σ)       (φσ)

Theoremsyl213anc 1153 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (φζ)    &   (((ψ χ) θ (τ η ζ)) → σ)       (φσ)

Theoremsyl231anc 1154 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (φζ)    &   (((ψ χ) (θ τ η) ζ) → σ)       (φσ)

Theoremsyl312anc 1155 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (φζ)    &   (((ψ χ θ) τ (η ζ)) → σ)       (φσ)

Theoremsyl321anc 1156 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (φζ)    &   (((ψ χ θ) (τ η) ζ) → σ)       (φσ)

Theoremsyl133anc 1157 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (φζ)    &   (φσ)    &   ((ψ (χ θ τ) (η ζ σ)) → ρ)       (φρ)

Theoremsyl313anc 1158 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (φζ)    &   (φσ)    &   (((ψ χ θ) τ (η ζ σ)) → ρ)       (φρ)

Theoremsyl331anc 1159 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (φζ)    &   (φσ)    &   (((ψ χ θ) (τ η ζ) σ) → ρ)       (φρ)

Theoremsyl223anc 1160 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (φζ)    &   (φσ)    &   (((ψ χ) (θ τ) (η ζ σ)) → ρ)       (φρ)

Theoremsyl232anc 1161 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (φζ)    &   (φσ)    &   (((ψ χ) (θ τ η) (ζ σ)) → ρ)       (φρ)

Theoremsyl322anc 1162 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (φζ)    &   (φσ)    &   (((ψ χ θ) (τ η) (ζ σ)) → ρ)       (φρ)

Theoremsyl233anc 1163 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (φζ)    &   (φσ)    &   (φρ)    &   (((ψ χ) (θ τ η) (ζ σ ρ)) → μ)       (φμ)

Theoremsyl323anc 1164 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (φζ)    &   (φσ)    &   (φρ)    &   (((ψ χ θ) (τ η) (ζ σ ρ)) → μ)       (φμ)

Theoremsyl332anc 1165 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (φζ)    &   (φσ)    &   (φρ)    &   (((ψ χ θ) (τ η ζ) (σ ρ)) → μ)       (φμ)

Theoremsyl333anc 1166 A syllogism inference combined with contraction. (Contributed by NM, 10-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (φζ)    &   (φσ)    &   (φρ)    &   (φμ)    &   (((ψ χ θ) (τ η ζ) (σ ρ μ)) → λ)       (φλ)

Theoremsyl3an1 1167 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(φψ)    &   ((ψ χ θ) → τ)       ((φ χ θ) → τ)

Theoremsyl3an2 1168 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(φχ)    &   ((ψ χ θ) → τ)       ((ψ φ θ) → τ)

Theoremsyl3an3 1169 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(φθ)    &   ((ψ χ θ) → τ)       ((ψ χ φ) → τ)

Theoremsyl3an1b 1170 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(φψ)    &   ((ψ χ θ) → τ)       ((φ χ θ) → τ)

Theoremsyl3an2b 1171 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(φχ)    &   ((ψ χ θ) → τ)       ((ψ φ θ) → τ)

Theoremsyl3an3b 1172 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(φθ)    &   ((ψ χ θ) → τ)       ((ψ χ φ) → τ)

Theoremsyl3an1br 1173 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(ψφ)    &   ((ψ χ θ) → τ)       ((φ χ θ) → τ)

Theoremsyl3an2br 1174 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(χφ)    &   ((ψ χ θ) → τ)       ((ψ φ θ) → τ)

Theoremsyl3an3br 1175 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(θφ)    &   ((ψ χ θ) → τ)       ((ψ χ φ) → τ)

Theoremsyl3an 1176 A triple syllogism inference. (Contributed by NM, 13-May-2004.)
(φψ)    &   (χθ)    &   (τη)    &   ((ψ θ η) → ζ)       ((φ χ τ) → ζ)

Theoremsyl3anb 1177 A triple syllogism inference. (Contributed by NM, 15-Oct-2005.)
(φψ)    &   (χθ)    &   (τη)    &   ((ψ θ η) → ζ)       ((φ χ τ) → ζ)

Theoremsyl3anbr 1178 A triple syllogism inference. (Contributed by NM, 29-Dec-2011.)
(ψφ)    &   (θχ)    &   (ητ)    &   ((ψ θ η) → ζ)       ((φ χ τ) → ζ)

Theoremsyld3an3 1179 A syllogism inference. (Contributed by NM, 20-May-2007.)
((φ ψ χ) → θ)    &   ((φ ψ θ) → τ)       ((φ ψ χ) → τ)

Theoremsyld3an1 1180 A syllogism inference. (Contributed by NM, 7-Jul-2008.)
((χ ψ θ) → φ)    &   ((φ ψ θ) → τ)       ((χ ψ θ) → τ)

Theoremsyld3an2 1181 A syllogism inference. (Contributed by NM, 20-May-2007.)
((φ χ θ) → ψ)    &   ((φ ψ θ) → τ)       ((φ χ θ) → τ)

Theoremsyl3anl1 1182 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
(φψ)    &   (((ψ χ θ) τ) → η)       (((φ χ θ) τ) → η)

Theoremsyl3anl2 1183 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
(φχ)    &   (((ψ χ θ) τ) → η)       (((ψ φ θ) τ) → η)

Theoremsyl3anl3 1184 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
(φθ)    &   (((ψ χ θ) τ) → η)       (((ψ χ φ) τ) → η)

Theoremsyl3anl 1185 A triple syllogism inference. (Contributed by NM, 24-Dec-2006.)
(φψ)    &   (χθ)    &   (τη)    &   (((ψ θ η) ζ) → σ)       (((φ χ τ) ζ) → σ)

Theoremsyl3anr1 1186 A syllogism inference. (Contributed by NM, 31-Jul-2007.)
(φψ)    &   ((χ (ψ θ τ)) → η)       ((χ (φ θ τ)) → η)

Theoremsyl3anr2 1187 A syllogism inference. (Contributed by NM, 1-Aug-2007.)
(φθ)    &   ((χ (ψ θ τ)) → η)       ((χ (ψ φ τ)) → η)

Theoremsyl3anr3 1188 A syllogism inference. (Contributed by NM, 23-Aug-2007.)
(φτ)    &   ((χ (ψ θ τ)) → η)       ((χ (ψ θ φ)) → η)

Theorem3impdi 1189 Importation inference (undistribute conjunction). (Contributed by NM, 14-Aug-1995.)
(((φ ψ) (φ χ)) → θ)       ((φ ψ χ) → θ)

Theorem3impdir 1190 Importation inference (undistribute conjunction). (Contributed by NM, 20-Aug-1995.)
(((φ ψ) (χ ψ)) → θ)       ((φ χ ψ) → θ)

Theorem3anidm12 1191 Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
((φ φ ψ) → χ)       ((φ ψ) → χ)

Theorem3anidm13 1192 Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
((φ ψ φ) → χ)       ((φ ψ) → χ)

Theorem3anidm23 1193 Inference from idempotent law for conjunction. (Contributed by NM, 1-Feb-2007.)
((φ ψ ψ) → χ)       ((φ ψ) → χ)

Theorem3ori 1194 Infer implication from triple disjunction. (Contributed by NM, 26-Sep-2006.)
(φ ψ χ)       ((¬ φ ¬ ψ) → χ)

Theorem3jao 1195 Disjunction of 3 antecedents. (Contributed by NM, 8-Apr-1994.)
(((φψ) (χψ) (θψ)) → ((φ χ θ) → ψ))

Theorem3jaob 1196 Disjunction of 3 antecedents. (Contributed by NM, 13-Sep-2011.)
(((φ χ θ) → ψ) ↔ ((φψ) (χψ) (θψ)))

Theorem3jaoi 1197 Disjunction of 3 antecedents (inference). (Contributed by NM, 12-Sep-1995.)
(φψ)    &   (χψ)    &   (θψ)       ((φ χ θ) → ψ)

Theorem3jaod 1198 Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.)
(φ → (ψχ))    &   (φ → (θχ))    &   (φ → (τχ))       (φ → ((ψ θ τ) → χ))

Theorem3jaoian 1199 Disjunction of 3 antecedents (inference). (Contributed by NM, 14-Oct-2005.)
((φ ψ) → χ)    &   ((θ ψ) → χ)    &   ((τ ψ) → χ)       (((φ θ τ) ψ) → χ)

Theorem3jaodan 1200 Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.)
((φ ψ) → χ)    &   ((φ θ) → χ)    &   ((φ τ) → χ)       ((φ (ψ θ τ)) → χ)

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