Home Intuitionistic Logic ExplorerTheorem List (p. 13 of 135) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 1201-1300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem3imp2 1201 Importation to right triple conjunction. (Contributed by NM, 26-Oct-2006.)
(𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))       ((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)

Theorem3exp1 1202 Exportation from left triple conjunction. (Contributed by NM, 24-Feb-2005.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theorem3expd 1203 Exportation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.)
(𝜑 → ((𝜓𝜒𝜃) → 𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theorem3exp2 1204 Exportation from right triple conjunction. (Contributed by NM, 26-Oct-2006.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp5o 1205 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
((𝜑𝜓𝜒) → ((𝜃𝜏) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theoremexp516 1206 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(((𝜑 ∧ (𝜓𝜒𝜃)) ∧ 𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theoremexp520 1207 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(((𝜑𝜓𝜒) ∧ (𝜃𝜏)) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theorem3anassrs 1208 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by Mario Carneiro, 4-Jan-2017.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜏)       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theorem3adant1l 1209 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓𝜒) → 𝜃)       (((𝜏𝜑) ∧ 𝜓𝜒) → 𝜃)

Theorem3adant1r 1210 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓𝜒) → 𝜃)       (((𝜑𝜏) ∧ 𝜓𝜒) → 𝜃)

Theorem3adant2l 1211 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜏𝜓) ∧ 𝜒) → 𝜃)

Theorem3adant2r 1212 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑 ∧ (𝜓𝜏) ∧ 𝜒) → 𝜃)

Theorem3adant3l 1213 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓 ∧ (𝜏𝜒)) → 𝜃)

Theorem3adant3r 1214 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
((𝜑𝜓𝜒) → 𝜃)       ((𝜑𝜓 ∧ (𝜒𝜏)) → 𝜃)

Theoremsyl12anc 1215 Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   ((𝜓 ∧ (𝜒𝜃)) → 𝜏)       (𝜑𝜏)

Theoremsyl21anc 1216 Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (((𝜓𝜒) ∧ 𝜃) → 𝜏)       (𝜑𝜏)

Theoremsyl3anc 1217 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   ((𝜓𝜒𝜃) → 𝜏)       (𝜑𝜏)

Theoremsyl22anc 1218 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (((𝜓𝜒) ∧ (𝜃𝜏)) → 𝜂)       (𝜑𝜂)

Theoremsyl13anc 1219 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   ((𝜓 ∧ (𝜒𝜃𝜏)) → 𝜂)       (𝜑𝜂)

Theoremsyl31anc 1220 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)       (𝜑𝜂)

Theoremsyl112anc 1221 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   ((𝜓𝜒 ∧ (𝜃𝜏)) → 𝜂)       (𝜑𝜂)

Theoremsyl121anc 1222 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   ((𝜓 ∧ (𝜒𝜃) ∧ 𝜏) → 𝜂)       (𝜑𝜂)

Theoremsyl211anc 1223 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (((𝜓𝜒) ∧ 𝜃𝜏) → 𝜂)       (𝜑𝜂)

Theoremsyl23anc 1224 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((𝜓𝜒) ∧ (𝜃𝜏𝜂)) → 𝜁)       (𝜑𝜁)

Theoremsyl32anc 1225 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂)) → 𝜁)       (𝜑𝜁)

Theoremsyl122anc 1226 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   ((𝜓 ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → 𝜁)       (𝜑𝜁)

Theoremsyl212anc 1227 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((𝜓𝜒) ∧ 𝜃 ∧ (𝜏𝜂)) → 𝜁)       (𝜑𝜁)

Theoremsyl221anc 1228 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((𝜓𝜒) ∧ (𝜃𝜏) ∧ 𝜂) → 𝜁)       (𝜑𝜁)

Theoremsyl113anc 1229 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   ((𝜓𝜒 ∧ (𝜃𝜏𝜂)) → 𝜁)       (𝜑𝜁)

Theoremsyl131anc 1230 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   ((𝜓 ∧ (𝜒𝜃𝜏) ∧ 𝜂) → 𝜁)       (𝜑𝜁)

Theoremsyl311anc 1231 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (((𝜓𝜒𝜃) ∧ 𝜏𝜂) → 𝜁)       (𝜑𝜁)

Theoremsyl33anc 1232 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁)) → 𝜎)       (𝜑𝜎)

Theoremsyl222anc 1233 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒) ∧ (𝜃𝜏) ∧ (𝜂𝜁)) → 𝜎)       (𝜑𝜎)

Theoremsyl123anc 1234 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   ((𝜓 ∧ (𝜒𝜃) ∧ (𝜏𝜂𝜁)) → 𝜎)       (𝜑𝜎)

Theoremsyl132anc 1235 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   ((𝜓 ∧ (𝜒𝜃𝜏) ∧ (𝜂𝜁)) → 𝜎)       (𝜑𝜎)

Theoremsyl213anc 1236 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒) ∧ 𝜃 ∧ (𝜏𝜂𝜁)) → 𝜎)       (𝜑𝜎)

Theoremsyl231anc 1237 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ 𝜁) → 𝜎)       (𝜑𝜎)

Theoremsyl312anc 1238 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒𝜃) ∧ 𝜏 ∧ (𝜂𝜁)) → 𝜎)       (𝜑𝜎)

Theoremsyl321anc 1239 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂) ∧ 𝜁) → 𝜎)       (𝜑𝜎)

Theoremsyl133anc 1240 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   ((𝜓 ∧ (𝜒𝜃𝜏) ∧ (𝜂𝜁𝜎)) → 𝜌)       (𝜑𝜌)

Theoremsyl313anc 1241 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒𝜃) ∧ 𝜏 ∧ (𝜂𝜁𝜎)) → 𝜌)       (𝜑𝜌)

Theoremsyl331anc 1242 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ 𝜎) → 𝜌)       (𝜑𝜌)

Theoremsyl223anc 1243 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒) ∧ (𝜃𝜏) ∧ (𝜂𝜁𝜎)) → 𝜌)       (𝜑𝜌)

Theoremsyl232anc 1244 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ (𝜁𝜎)) → 𝜌)       (𝜑𝜌)

Theoremsyl322anc 1245 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂) ∧ (𝜁𝜎)) → 𝜌)       (𝜑𝜌)

Theoremsyl233anc 1246 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (𝜑𝜌)    &   (((𝜓𝜒) ∧ (𝜃𝜏𝜂) ∧ (𝜁𝜎𝜌)) → 𝜇)       (𝜑𝜇)

Theoremsyl323anc 1247 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (𝜑𝜌)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂) ∧ (𝜁𝜎𝜌)) → 𝜇)       (𝜑𝜇)

Theoremsyl332anc 1248 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (𝜑𝜌)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ (𝜎𝜌)) → 𝜇)       (𝜑𝜇)

Theoremsyl333anc 1249 A syllogism inference combined with contraction. (Contributed by NM, 10-Mar-2012.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜑𝜂)    &   (𝜑𝜁)    &   (𝜑𝜎)    &   (𝜑𝜌)    &   (𝜑𝜇)    &   (((𝜓𝜒𝜃) ∧ (𝜏𝜂𝜁) ∧ (𝜎𝜌𝜇)) → 𝜆)       (𝜑𝜆)

Theoremsyl3an1 1250 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜓)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜑𝜒𝜃) → 𝜏)

Theoremsyl3an2 1251 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜒)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜑𝜃) → 𝜏)

Theoremsyl3an3 1252 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜃)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜒𝜑) → 𝜏)

Theoremsyl3an1b 1253 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜓)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜑𝜒𝜃) → 𝜏)

Theoremsyl3an2b 1254 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜒)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜑𝜃) → 𝜏)

Theoremsyl3an3b 1255 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜑𝜃)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜒𝜑) → 𝜏)

Theoremsyl3an1br 1256 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜓𝜑)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜑𝜒𝜃) → 𝜏)

Theoremsyl3an2br 1257 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜒𝜑)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜑𝜃) → 𝜏)

Theoremsyl3an3br 1258 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
(𝜃𝜑)    &   ((𝜓𝜒𝜃) → 𝜏)       ((𝜓𝜒𝜑) → 𝜏)

Theoremsyl3an 1259 A triple syllogism inference. (Contributed by NM, 13-May-2004.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)    &   ((𝜓𝜃𝜂) → 𝜁)       ((𝜑𝜒𝜏) → 𝜁)

Theoremsyl3anb 1260 A triple syllogism inference. (Contributed by NM, 15-Oct-2005.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)    &   ((𝜓𝜃𝜂) → 𝜁)       ((𝜑𝜒𝜏) → 𝜁)

Theoremsyl3anbr 1261 A triple syllogism inference. (Contributed by NM, 29-Dec-2011.)
(𝜓𝜑)    &   (𝜃𝜒)    &   (𝜂𝜏)    &   ((𝜓𝜃𝜂) → 𝜁)       ((𝜑𝜒𝜏) → 𝜁)

Theoremsyld3an3 1262 A syllogism inference. (Contributed by NM, 20-May-2007.)
((𝜑𝜓𝜒) → 𝜃)    &   ((𝜑𝜓𝜃) → 𝜏)       ((𝜑𝜓𝜒) → 𝜏)

Theoremsyld3an1 1263 A syllogism inference. (Contributed by NM, 7-Jul-2008.)
((𝜒𝜓𝜃) → 𝜑)    &   ((𝜑𝜓𝜃) → 𝜏)       ((𝜒𝜓𝜃) → 𝜏)

Theoremsyld3an2 1264 A syllogism inference. (Contributed by NM, 20-May-2007.)
((𝜑𝜒𝜃) → 𝜓)    &   ((𝜑𝜓𝜃) → 𝜏)       ((𝜑𝜒𝜃) → 𝜏)

Theoremsyl3anl1 1265 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
(𝜑𝜓)    &   (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)       (((𝜑𝜒𝜃) ∧ 𝜏) → 𝜂)

Theoremsyl3anl2 1266 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
(𝜑𝜒)    &   (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)       (((𝜓𝜑𝜃) ∧ 𝜏) → 𝜂)

Theoremsyl3anl3 1267 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
(𝜑𝜃)    &   (((𝜓𝜒𝜃) ∧ 𝜏) → 𝜂)       (((𝜓𝜒𝜑) ∧ 𝜏) → 𝜂)

Theoremsyl3anl 1268 A triple syllogism inference. (Contributed by NM, 24-Dec-2006.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜏𝜂)    &   (((𝜓𝜃𝜂) ∧ 𝜁) → 𝜎)       (((𝜑𝜒𝜏) ∧ 𝜁) → 𝜎)

Theoremsyl3anr1 1269 A syllogism inference. (Contributed by NM, 31-Jul-2007.)
(𝜑𝜓)    &   ((𝜒 ∧ (𝜓𝜃𝜏)) → 𝜂)       ((𝜒 ∧ (𝜑𝜃𝜏)) → 𝜂)

Theoremsyl3anr2 1270 A syllogism inference. (Contributed by NM, 1-Aug-2007.)
(𝜑𝜃)    &   ((𝜒 ∧ (𝜓𝜃𝜏)) → 𝜂)       ((𝜒 ∧ (𝜓𝜑𝜏)) → 𝜂)

Theoremsyl3anr3 1271 A syllogism inference. (Contributed by NM, 23-Aug-2007.)
(𝜑𝜏)    &   ((𝜒 ∧ (𝜓𝜃𝜏)) → 𝜂)       ((𝜒 ∧ (𝜓𝜃𝜑)) → 𝜂)

Theorem3impdi 1272 Importation inference (undistribute conjunction). (Contributed by NM, 14-Aug-1995.)
(((𝜑𝜓) ∧ (𝜑𝜒)) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)

Theorem3impdir 1273 Importation inference (undistribute conjunction). (Contributed by NM, 20-Aug-1995.)
(((𝜑𝜓) ∧ (𝜒𝜓)) → 𝜃)       ((𝜑𝜒𝜓) → 𝜃)

Theorem3anidm12 1274 Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
((𝜑𝜑𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theorem3anidm13 1275 Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
((𝜑𝜓𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theorem3anidm23 1276 Inference from idempotent law for conjunction. (Contributed by NM, 1-Feb-2007.)
((𝜑𝜓𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theoremsyl2an3an 1277 syl3an 1259 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜃𝜏)    &   ((𝜓𝜒𝜏) → 𝜂)       ((𝜑𝜃) → 𝜂)

Theoremsyl2an23an 1278 Deduction related to syl3an 1259 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.)
(𝜑𝜓)    &   (𝜑𝜒)    &   ((𝜃𝜑) → 𝜏)    &   ((𝜓𝜒𝜏) → 𝜂)       ((𝜃𝜑) → 𝜂)

Theorem3ori 1279 Infer implication from triple disjunction. (Contributed by NM, 26-Sep-2006.)
(𝜑𝜓𝜒)       ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)

Theorem3jao 1280 Disjunction of 3 antecedents. (Contributed by NM, 8-Apr-1994.)
(((𝜑𝜓) ∧ (𝜒𝜓) ∧ (𝜃𝜓)) → ((𝜑𝜒𝜃) → 𝜓))

Theorem3jaob 1281 Disjunction of 3 antecedents. (Contributed by NM, 13-Sep-2011.)
(((𝜑𝜒𝜃) → 𝜓) ↔ ((𝜑𝜓) ∧ (𝜒𝜓) ∧ (𝜃𝜓)))

Theorem3jaoi 1282 Disjunction of 3 antecedents (inference). (Contributed by NM, 12-Sep-1995.)
(𝜑𝜓)    &   (𝜒𝜓)    &   (𝜃𝜓)       ((𝜑𝜒𝜃) → 𝜓)

Theorem3jaod 1283 Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜒))    &   (𝜑 → (𝜏𝜒))       (𝜑 → ((𝜓𝜃𝜏) → 𝜒))

Theorem3jaoian 1284 Disjunction of 3 antecedents (inference). (Contributed by NM, 14-Oct-2005.)
((𝜑𝜓) → 𝜒)    &   ((𝜃𝜓) → 𝜒)    &   ((𝜏𝜓) → 𝜒)       (((𝜑𝜃𝜏) ∧ 𝜓) → 𝜒)

Theorem3jaodan 1285 Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜒)    &   ((𝜑𝜏) → 𝜒)       ((𝜑 ∧ (𝜓𝜃𝜏)) → 𝜒)

Theoremmpjao3dan 1286 Eliminate a 3-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜃) → 𝜒)    &   ((𝜑𝜏) → 𝜒)    &   (𝜑 → (𝜓𝜃𝜏))       (𝜑𝜒)

Theorem3jaao 1287 Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜒))    &   (𝜂 → (𝜁𝜒))       ((𝜑𝜃𝜂) → ((𝜓𝜏𝜁) → 𝜒))

Theorem3ianorr 1288 Triple disjunction implies negated triple conjunction. (Contributed by Jim Kingdon, 23-Dec-2018.)
((¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒) → ¬ (𝜑𝜓𝜒))

Theoremsyl3an9b 1289 Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜒𝜏))    &   (𝜂 → (𝜏𝜁))       ((𝜑𝜃𝜂) → (𝜓𝜁))

Theorem3orbi123d 1290 Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → (𝜂𝜁))       (𝜑 → ((𝜓𝜃𝜂) ↔ (𝜒𝜏𝜁)))

Theorem3anbi123d 1291 Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → (𝜂𝜁))       (𝜑 → ((𝜓𝜃𝜂) ↔ (𝜒𝜏𝜁)))

Theorem3anbi12d 1292 Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃𝜂) ↔ (𝜒𝜏𝜂)))

Theorem3anbi13d 1293 Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜂𝜃) ↔ (𝜒𝜂𝜏)))

Theorem3anbi23d 1294 Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜂𝜓𝜃) ↔ (𝜂𝜒𝜏)))

Theorem3anbi1d 1295 Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃𝜏) ↔ (𝜒𝜃𝜏)))

Theorem3anbi2d 1296 Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓𝜏) ↔ (𝜃𝜒𝜏)))

Theorem3anbi3d 1297 Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜏𝜓) ↔ (𝜃𝜏𝜒)))

Theorem3anim123d 1298 Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → (𝜂𝜁))       (𝜑 → ((𝜓𝜃𝜂) → (𝜒𝜏𝜁)))

Theorem3orim123d 1299 Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜑 → (𝜂𝜁))       (𝜑 → ((𝜓𝜃𝜂) → (𝜒𝜏𝜁)))

Theoreman6 1300 Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.)
(((𝜑𝜓𝜒) ∧ (𝜃𝜏𝜂)) ↔ ((𝜑𝜃) ∧ (𝜓𝜏) ∧ (𝜒𝜂)))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13470
 Copyright terms: Public domain < Previous  Next >