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Theorem List for New Foundations Explorer - 1101-1200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsimp233 1101 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η (θ τ (φ ψ χ)) ζ) → χ)
 
Theoremsimp311 1102 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ ((φ ψ χ) θ τ)) → φ)
 
Theoremsimp312 1103 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ ((φ ψ χ) θ τ)) → ψ)
 
Theoremsimp313 1104 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ ((φ ψ χ) θ τ)) → χ)
 
Theoremsimp321 1105 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ (θ (φ ψ χ) τ)) → φ)
 
Theoremsimp322 1106 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ (θ (φ ψ χ) τ)) → ψ)
 
Theoremsimp323 1107 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ (θ (φ ψ χ) τ)) → χ)
 
Theoremsimp331 1108 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ (θ τ (φ ψ χ))) → φ)
 
Theoremsimp332 1109 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ (θ τ (φ ψ χ))) → ψ)
 
Theoremsimp333 1110 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((η ζ (θ τ (φ ψ χ))) → χ)
 
Theorem3adantl1 1111 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((φ ψ) χ) → θ)       (((τ φ ψ) χ) → θ)
 
Theorem3adantl2 1112 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((φ ψ) χ) → θ)       (((φ τ ψ) χ) → θ)
 
Theorem3adantl3 1113 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
(((φ ψ) χ) → θ)       (((φ ψ τ) χ) → θ)
 
Theorem3adantr1 1114 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((φ (ψ χ)) → θ)       ((φ (τ ψ χ)) → θ)
 
Theorem3adantr2 1115 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((φ (ψ χ)) → θ)       ((φ (ψ τ χ)) → θ)
 
Theorem3adantr3 1116 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
((φ (ψ χ)) → θ)       ((φ (ψ χ τ)) → θ)
 
Theorem3ad2antl1 1117 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((φ χ) → θ)       (((φ ψ τ) χ) → θ)
 
Theorem3ad2antl2 1118 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((φ χ) → θ)       (((ψ φ τ) χ) → θ)
 
Theorem3ad2antl3 1119 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
((φ χ) → θ)       (((ψ τ φ) χ) → θ)
 
Theorem3ad2antr1 1120 Deduction adding conjuncts to antecedent. (Contributed by NM, 25-Dec-2007.)
((φ χ) → θ)       ((φ (χ ψ τ)) → θ)
 
Theorem3ad2antr2 1121 Deduction adding conjuncts to antecedent. (Contributed by NM, 27-Dec-2007.)
((φ χ) → θ)       ((φ (ψ χ τ)) → θ)
 
Theorem3ad2antr3 1122 Deduction adding conjuncts to antecedent. (Contributed by NM, 30-Dec-2007.)
((φ χ) → θ)       ((φ (ψ τ χ)) → θ)
 
Theorem3anibar 1123 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 1124 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(φ → (φ ψ χ))
 
Theorem3mix2 1125 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(φ → (ψ φ χ))
 
Theorem3mix3 1126 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
(φ → (ψ χ φ))
 
Theorem3mix1i 1127 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
φ       (φ ψ χ)
 
Theorem3mix2i 1128 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
φ       (ψ φ χ)
 
Theorem3mix3i 1129 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
φ       (ψ χ φ)
 
Theorem3pm3.2i 1130 Infer conjunction of premises. (Contributed by NM, 10-Feb-1995.)
φ    &   ψ    &   χ       (φ ψ χ)
 
Theorempm3.2an3 1131 pm3.2 434 for a triple conjunction. (Contributed by Alan Sare, 24-Oct-2011.)
(φ → (ψ → (χ → (φ ψ χ))))
 
Theorem3jca 1132 Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.)
(φψ)    &   (φχ)    &   (φθ)       (φ → (ψ χ θ))
 
Theorem3jcad 1133 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 1134 Detach a conjunction of truths in a biconditional. (Contributed by NM, 16-Sep-2011.)
ψ    &   χ    &   θ    &   (φ ↔ (ψ χ θ))       φ
 
Theoremmpbir3and 1135 Detach a conjunction of truths in a biconditional. (Contributed by Mario Carneiro, 11-May-2014.) (Revised by Mario Carneiro, 9-Jan-2015.)
(φχ)    &   (φθ)    &   (φτ)    &   (φ → (ψ ↔ (χ θ τ)))       (φψ)
 
Theoremsyl3anbrc 1136 Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014.)
(φψ)    &   (φχ)    &   (φθ)    &   (τ ↔ (ψ χ θ))       (φτ)
 
Theorem3anim123i 1137 Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.)
(φψ)    &   (χθ)    &   (τη)       ((φ χ τ) → (ψ θ η))
 
Theorem3anim1i 1138 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 16-Aug-2009.)
(φψ)       ((φ χ θ) → (ψ χ θ))
 
Theorem3anim3i 1139 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 19-Aug-2009.)
(φψ)       ((χ θ φ) → (χ θ ψ))
 
Theorem3anbi123i 1140 Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994.)
(φψ)    &   (χθ)    &   (τη)       ((φ χ τ) ↔ (ψ θ η))
 
Theorem3orbi123i 1141 Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)
(φψ)    &   (χθ)    &   (τη)       ((φ χ τ) ↔ (ψ θ η))
 
Theorem3anbi1i 1142 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(φψ)       ((φ χ θ) ↔ (ψ χ θ))
 
Theorem3anbi2i 1143 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(φψ)       ((χ φ θ) ↔ (χ ψ θ))
 
Theorem3anbi3i 1144 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
(φψ)       ((χ θ φ) ↔ (χ θ ψ))
 
Theorem3imp 1145 Importation inference. (Contributed by NM, 8-Apr-1994.)
(φ → (ψ → (χθ)))       ((φ ψ χ) → θ)
 
Theorem3impa 1146 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
(((φ ψ) χ) → θ)       ((φ ψ χ) → θ)
 
Theorem3impb 1147 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
((φ (ψ χ)) → θ)       ((φ ψ χ) → θ)
 
Theorem3impia 1148 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
((φ ψ) → (χθ))       ((φ ψ χ) → θ)
 
Theorem3impib 1149 Importation to triple conjunction. (Contributed by NM, 13-Jun-2006.)
(φ → ((ψ χ) → θ))       ((φ ψ χ) → θ)
 
Theorem3exp 1150 Exportation inference. (Contributed by NM, 30-May-1994.)
((φ ψ χ) → θ)       (φ → (ψ → (χθ)))
 
Theorem3expa 1151 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
((φ ψ χ) → θ)       (((φ ψ) χ) → θ)
 
Theorem3expb 1152 Exportation from triple to double conjunction. (Contributed by NM, 20-Aug-1995.)
((φ ψ χ) → θ)       ((φ (ψ χ)) → θ)
 
Theorem3expia 1153 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
((φ ψ χ) → θ)       ((φ ψ) → (χθ))
 
Theorem3expib 1154 Exportation from triple conjunction. (Contributed by NM, 19-May-2007.)
((φ ψ χ) → θ)       (φ → ((ψ χ) → θ))
 
Theorem3com12 1155 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 1156 Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM, 28-Jan-1996.)
((φ ψ χ) → θ)       ((χ ψ φ) → θ)
 
Theorem3com23 1157 Commutation in antecedent. Swap 2nd and 3rd. (Contributed by NM, 28-Jan-1996.)
((φ ψ χ) → θ)       ((φ χ ψ) → θ)
 
Theorem3coml 1158 Commutation in antecedent. Rotate left. (Contributed by NM, 28-Jan-1996.)
((φ ψ χ) → θ)       ((ψ χ φ) → θ)
 
Theorem3comr 1159 Commutation in antecedent. Rotate right. (Contributed by NM, 28-Jan-1996.)
((φ ψ χ) → θ)       ((χ φ ψ) → θ)
 
Theorem3adant3r1 1160 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Feb-2008.)
((φ ψ χ) → θ)       ((φ (τ ψ χ)) → θ)
 
Theorem3adant3r2 1161 Deduction adding a conjunct to antecedent. (Contributed by NM, 17-Feb-2008.)
((φ ψ χ) → θ)       ((φ (ψ τ χ)) → θ)
 
Theorem3adant3r3 1162 Deduction adding a conjunct to antecedent. (Contributed by NM, 18-Feb-2008.)
((φ ψ χ) → θ)       ((φ (ψ χ τ)) → θ)
 
Theorem3an1rs 1163 Swap conjuncts. (Contributed by NM, 16-Dec-2007.)
(((φ ψ χ) θ) → τ)       (((φ ψ θ) χ) → τ)
 
Theorem3imp1 1164 Importation to left triple conjunction. (Contributed by NM, 24-Feb-2005.)
(φ → (ψ → (χ → (θτ))))       (((φ ψ χ) θ) → τ)
 
Theorem3impd 1165 Importation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.)
(φ → (ψ → (χ → (θτ))))       (φ → ((ψ χ θ) → τ))
 
Theorem3imp2 1166 Importation to right triple conjunction. (Contributed by NM, 26-Oct-2006.)
(φ → (ψ → (χ → (θτ))))       ((φ (ψ χ θ)) → τ)
 
Theorem3exp1 1167 Exportation from left triple conjunction. (Contributed by NM, 24-Feb-2005.)
(((φ ψ χ) θ) → τ)       (φ → (ψ → (χ → (θτ))))
 
Theorem3expd 1168 Exportation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.)
(φ → ((ψ χ θ) → τ))       (φ → (ψ → (χ → (θτ))))
 
Theorem3exp2 1169 Exportation from right triple conjunction. (Contributed by NM, 26-Oct-2006.)
((φ (ψ χ θ)) → τ)       (φ → (ψ → (χ → (θτ))))
 
Theoremexp5o 1170 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
((φ ψ χ) → ((θ τ) → η))       (φ → (ψ → (χ → (θ → (τη)))))
 
Theoremexp516 1171 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(((φ (ψ χ θ)) τ) → η)       (φ → (ψ → (χ → (θ → (τη)))))
 
Theoremexp520 1172 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(((φ ψ χ) (θ τ)) → η)       (φ → (ψ → (χ → (θ → (τη)))))
 
Theorem3anassrs 1173 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 1174 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((φ ψ χ) → θ)       (((τ φ) ψ χ) → θ)
 
Theorem3adant1r 1175 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((φ ψ χ) → θ)       (((φ τ) ψ χ) → θ)
 
Theorem3adant2l 1176 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((φ ψ χ) → θ)       ((φ (τ ψ) χ) → θ)
 
Theorem3adant2r 1177 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((φ ψ χ) → θ)       ((φ (ψ τ) χ) → θ)
 
Theorem3adant3l 1178 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((φ ψ χ) → θ)       ((φ ψ (τ χ)) → θ)
 
Theorem3adant3r 1179 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((φ ψ χ) → θ)       ((φ ψ (χ τ)) → θ)
 
Theoremsyl12anc 1180 Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
(φψ)    &   (φχ)    &   (φθ)    &   ((ψ (χ θ)) → τ)       (φτ)
 
Theoremsyl21anc 1181 Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
(φψ)    &   (φχ)    &   (φθ)    &   (((ψ χ) θ) → τ)       (φτ)
 
Theoremsyl3anc 1182 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   ((ψ χ θ) → τ)       (φτ)
 
Theoremsyl22anc 1183 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (((ψ χ) (θ τ)) → η)       (φη)
 
Theoremsyl13anc 1184 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   ((ψ (χ θ τ)) → η)       (φη)
 
Theoremsyl31anc 1185 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (((ψ χ θ) τ) → η)       (φη)
 
Theoremsyl112anc 1186 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   ((ψ χ (θ τ)) → η)       (φη)
 
Theoremsyl121anc 1187 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   ((ψ (χ θ) τ) → η)       (φη)
 
Theoremsyl211anc 1188 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (((ψ χ) θ τ) → η)       (φη)
 
Theoremsyl23anc 1189 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (((ψ χ) (θ τ η)) → ζ)       (φζ)
 
Theoremsyl32anc 1190 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (((ψ χ θ) (τ η)) → ζ)       (φζ)
 
Theoremsyl122anc 1191 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   ((ψ (χ θ) (τ η)) → ζ)       (φζ)
 
Theoremsyl212anc 1192 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (((ψ χ) θ (τ η)) → ζ)       (φζ)
 
Theoremsyl221anc 1193 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (((ψ χ) (θ τ) η) → ζ)       (φζ)
 
Theoremsyl113anc 1194 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   ((ψ χ (θ τ η)) → ζ)       (φζ)
 
Theoremsyl131anc 1195 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   ((ψ (χ θ τ) η) → ζ)       (φζ)
 
Theoremsyl311anc 1196 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (((ψ χ θ) τ η) → ζ)       (φζ)
 
Theoremsyl33anc 1197 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 1198 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 1199 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 1200 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
(φψ)    &   (φχ)    &   (φθ)    &   (φτ)    &   (φη)    &   (φζ)    &   ((ψ (χ θ τ) (η ζ)) → σ)       (φσ)
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