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Theorem List for Intuitionistic Logic Explorer - 1101-1200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsimp133 1101 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜂𝜁) → 𝜒)
 
Theoremsimp211 1102 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜁) → 𝜑)
 
Theoremsimp212 1103 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜁) → 𝜓)
 
Theoremsimp213 1104 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏) ∧ 𝜁) → 𝜒)
 
Theoremsimp221 1105 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜁) → 𝜑)
 
Theoremsimp222 1106 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜁) → 𝜓)
 
Theoremsimp223 1107 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏) ∧ 𝜁) → 𝜒)
 
Theoremsimp231 1108 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜁) → 𝜑)
 
Theoremsimp232 1109 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜁) → 𝜓)
 
Theoremsimp233 1110 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒)) ∧ 𝜁) → 𝜒)
 
Theoremsimp311 1111 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜑)
 
Theoremsimp312 1112 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜓)
 
Theoremsimp313 1113 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ ((𝜑𝜓𝜒) ∧ 𝜃𝜏)) → 𝜒)
 
Theoremsimp321 1114 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜑)
 
Theoremsimp322 1115 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜓)
 
Theoremsimp323 1116 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃 ∧ (𝜑𝜓𝜒) ∧ 𝜏)) → 𝜒)
 
Theoremsimp331 1117 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜑)
 
Theoremsimp332 1118 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜓)
 
Theoremsimp333 1119 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜂𝜁 ∧ (𝜃𝜏 ∧ (𝜑𝜓𝜒))) → 𝜒)
 
Theorem3adantl1 1120 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜏𝜑𝜓) ∧ 𝜒) → 𝜃)
 
Theorem3adantl2 1121 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜑𝜏𝜓) ∧ 𝜒) → 𝜃)
 
Theorem3adantl3 1122 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       (((𝜑𝜓𝜏) ∧ 𝜒) → 𝜃)
 
Theorem3adantr1 1123 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜏𝜓𝜒)) → 𝜃)
 
Theorem3adantr2 1124 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜓𝜏𝜒)) → 𝜃)
 
Theorem3adantr3 1125 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑 ∧ (𝜓𝜒𝜏)) → 𝜃)
 
Theorem3ad2antl1 1126 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((𝜑𝜒) → 𝜃)       (((𝜑𝜓𝜏) ∧ 𝜒) → 𝜃)
 
Theorem3ad2antl2 1127 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((𝜑𝜒) → 𝜃)       (((𝜓𝜑𝜏) ∧ 𝜒) → 𝜃)
 
Theorem3ad2antl3 1128 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((𝜑𝜒) → 𝜃)       (((𝜓𝜏𝜑) ∧ 𝜒) → 𝜃)
 
Theorem3ad2antr1 1129 Deduction adding a conjuncts to antecedent. (Contributed by NM, 25-Dec-2007.)
((𝜑𝜒) → 𝜃)       ((𝜑 ∧ (𝜒𝜓𝜏)) → 𝜃)
 
Theorem3ad2antr2 1130 Deduction adding a conjuncts to antecedent. (Contributed by NM, 27-Dec-2007.)
((𝜑𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒𝜏)) → 𝜃)
 
Theorem3ad2antr3 1131 Deduction adding a conjuncts to antecedent. (Contributed by NM, 30-Dec-2007.)
((𝜑𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜏𝜒)) → 𝜃)
 
Theorem3anibar 1132 Remove a hypothesis from the second member of a biimplication. (Contributed by FL, 22-Jul-2008.)
((𝜑𝜓𝜒) → (𝜃 ↔ (𝜒𝜏)))       ((𝜑𝜓𝜒) → (𝜃𝜏))
 
Theorem3mix1 1133 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(𝜑 → (𝜑𝜓𝜒))
 
Theorem3mix2 1134 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(𝜑 → (𝜓𝜑𝜒))
 
Theorem3mix3 1135 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(𝜑 → (𝜓𝜒𝜑))
 
Theorem3mix1i 1136 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
𝜑       (𝜑𝜓𝜒)
 
Theorem3mix2i 1137 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
𝜑       (𝜓𝜑𝜒)
 
Theorem3mix3i 1138 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
𝜑       (𝜓𝜒𝜑)
 
Theorem3mix1d 1139 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
(𝜑𝜓)       (𝜑 → (𝜓𝜒𝜃))
 
Theorem3mix2d 1140 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
(𝜑𝜓)       (𝜑 → (𝜒𝜓𝜃))
 
Theorem3mix3d 1141 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
(𝜑𝜓)       (𝜑 → (𝜒𝜃𝜓))
 
Theorem3pm3.2i 1142 Infer conjunction of premises. (Contributed by NM, 10-Feb-1995.)
𝜑    &   𝜓    &   𝜒       (𝜑𝜓𝜒)
 
Theorempm3.2an3 1143 pm3.2 138 for a triple conjunction. (Contributed by Alan Sare, 24-Oct-2011.)
(𝜑 → (𝜓 → (𝜒 → (𝜑𝜓𝜒))))
 
Theorem3jca 1144 Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)       (𝜑 → (𝜓𝜒𝜃))
 
Theorem3jcad 1145 Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜓𝜏))       (𝜑 → (𝜓 → (𝜒𝜃𝜏)))
 
Theoremmpbir3an 1146 Detach a conjunction of truths in a biconditional. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 9-Jan-2015.)
𝜓    &   𝜒    &   𝜃    &   (𝜑 ↔ (𝜓𝜒𝜃))       𝜑
 
Theoremmpbir3and 1147 Detach a conjunction of truths in a biconditional. (Contributed by Mario Carneiro, 11-May-2014.)
(𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑 → (𝜓 ↔ (𝜒𝜃𝜏)))       (𝜑𝜓)
 
Theoremsyl3anbrc 1148 Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜏 ↔ (𝜓𝜒𝜃))       (𝜑𝜏)
 
Theorem3anim123i 1149 Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       ((𝜑𝜒𝜏) → (𝜓𝜃𝜂))
 
Theorem3anim1i 1150 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 16-Aug-2009.)
(𝜑𝜓)       ((𝜑𝜒𝜃) → (𝜓𝜒𝜃))
 
Theorem3anim2i 1151 Add two conjuncts to antecedent and consequent. (Contributed by AV, 21-Nov-2019.)
(𝜑𝜓)       ((𝜒𝜑𝜃) → (𝜒𝜓𝜃))
 
Theorem3anim3i 1152 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 19-Aug-2009.)
(𝜑𝜓)       ((𝜒𝜃𝜑) → (𝜒𝜃𝜓))
 
Theorem3anbi123i 1153 Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))
 
Theorem3orbi123i 1154 Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)       ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂))
 
Theorem3anbi1i 1155 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(𝜑𝜓)       ((𝜑𝜒𝜃) ↔ (𝜓𝜒𝜃))
 
Theorem3anbi2i 1156 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(𝜑𝜓)       ((𝜒𝜑𝜃) ↔ (𝜒𝜓𝜃))
 
Theorem3anbi3i 1157 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(𝜑𝜓)       ((𝜒𝜃𝜑) ↔ (𝜒𝜃𝜓))
 
Theorem3imp 1158 Importation inference. (Contributed by NM, 8-Apr-1994.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3impa 1159 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
(((𝜑𝜓) ∧ 𝜒) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3impb 1160 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3impia 1161 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
((𝜑𝜓) → (𝜒𝜃))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3impib 1162 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
(𝜑 → ((𝜓𝜒) → 𝜃))       ((𝜑𝜓𝜒) → 𝜃)
 
Theorem3exp 1163 Exportation inference. (Contributed by NM, 30-May-1994.)
((𝜑𝜓𝜒) → 𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theorem3expa 1164 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
((𝜑𝜓𝜒) → 𝜃)       (((𝜑𝜓) ∧ 𝜒) → 𝜃)
 
Theorem3expb 1165 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
 
Theorem3expia 1166 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓) → (𝜒𝜃))
 
Theorem3expib 1167 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
((𝜑𝜓𝜒) → 𝜃)       (𝜑 → ((𝜓𝜒) → 𝜃))
 
Theorem3com12 1168 Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM, 28-Jan-1996.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜓𝜑𝜒) → 𝜃)
 
Theorem3com13 1169 Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM, 28-Jan-1996.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜒𝜓𝜑) → 𝜃)
 
Theorem3com23 1170 Commutation in antecedent. Swap 2nd and 3rd. (Contributed by NM, 28-Jan-1996.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜒𝜓) → 𝜃)
 
Theorem3coml 1171 Commutation in antecedent. Rotate left. (Contributed by NM, 28-Jan-1996.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜓𝜒𝜑) → 𝜃)
 
Theorem3comr 1172 Commutation in antecedent. Rotate right. (Contributed by NM, 28-Jan-1996.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜒𝜑𝜓) → 𝜃)
 
Theorem3adant3r1 1173 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Feb-2008.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜏𝜓𝜒)) → 𝜃)
 
Theorem3adant3r2 1174 Deduction adding a conjunct to antecedent. (Contributed by NM, 17-Feb-2008.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜏𝜒)) → 𝜃)
 
Theorem3adant3r3 1175 Deduction adding a conjunct to antecedent. (Contributed by NM, 18-Feb-2008.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜒𝜏)) → 𝜃)
 
Theoremad4ant123 1176 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓𝜒) → 𝜃)       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜃)
 
Theoremad4ant124 1177 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓𝜒) → 𝜃)       ((((𝜑𝜓) ∧ 𝜏) ∧ 𝜒) → 𝜃)
 
Theoremad4ant134 1178 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓𝜒) → 𝜃)       ((((𝜑𝜏) ∧ 𝜓) ∧ 𝜒) → 𝜃)
 
Theoremad4ant234 1179 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
((𝜑𝜓𝜒) → 𝜃)       ((((𝜏𝜑) ∧ 𝜓) ∧ 𝜒) → 𝜃)
 
Theorem3an1rs 1180 Swap conjuncts. (Contributed by NM, 16-Dec-2007.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (((𝜑𝜓𝜃) ∧ 𝜒) → 𝜏)
 
Theorem3imp1 1181 Importation to left triple conjunction. (Contributed by NM, 24-Feb-2005.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)
 
Theorem3impd 1182 Importation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       (𝜑 → ((𝜓𝜒𝜃) → 𝜏))
 
Theorem3imp2 1183 Importation to right triple conjunction. (Contributed by NM, 26-Oct-2006.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)
 
Theorem3exp1 1184 Exportation from left triple conjunction. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theorem3expd 1185 Exportation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.)
(𝜑 → ((𝜓𝜒𝜃) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theorem3exp2 1186 Exportation from right triple conjunction. (Contributed by NM, 26-Oct-2006.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 
Theoremexp5o 1187 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
((𝜑𝜓𝜒) → ((𝜃𝜏) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp516 1188 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(((𝜑 ∧ (𝜓𝜒𝜃)) ∧ 𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theoremexp520 1189 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(((𝜑𝜓𝜒) ∧ (𝜃𝜏)) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))
 
Theorem3anassrs 1190 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by Mario Carneiro, 4-Jan-2017.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
 
Theorem3adant1l 1191 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓𝜒) → 𝜃)       (((𝜏𝜑) ∧ 𝜓𝜒) → 𝜃)
 
Theorem3adant1r 1192 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓𝜒) → 𝜃)       (((𝜑𝜏) ∧ 𝜓𝜒) → 𝜃)
 
Theorem3adant2l 1193 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜏𝜓) ∧ 𝜒) → 𝜃)
 
Theorem3adant2r 1194 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜏) ∧ 𝜒) → 𝜃)
 
Theorem3adant3l 1195 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓 ∧ (𝜏𝜒)) → 𝜃)
 
Theorem3adant3r 1196 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓 ∧ (𝜒𝜏)) → 𝜃)
 
Theoremsyl12anc 1197 Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   ((𝜓 ∧ (𝜒𝜃)) → 𝜏)       (𝜑𝜏)
 
Theoremsyl21anc 1198 Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (((𝜓𝜒) ∧ 𝜃) → 𝜏)       (𝜑𝜏)
 
Theoremsyl3anc 1199 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   ((𝜓𝜒𝜃) → 𝜏)       (𝜑𝜏)
 
Theoremsyl22anc 1200 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (((𝜓𝜒) ∧ (𝜃𝜏)) → 𝜂)       (𝜑𝜂)
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