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Theorem List for Intuitionistic Logic Explorer - 1201-1300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem3adant3r1 1201 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Feb-2008.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ( ta 
 /\  ps  /\  ch )
 )  ->  th )
 
Theorem3adant3r2 1202 Deduction adding a conjunct to antecedent. (Contributed by NM, 17-Feb-2008.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ( ps 
 /\  ta  /\  ch )
 )  ->  th )
 
Theorem3adant3r3 1203 Deduction adding a conjunct to antecedent. (Contributed by NM, 18-Feb-2008.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ( ps 
 /\  ch  /\  ta )
 )  ->  th )
 
Theoremad4ant123 1204 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  ta )  ->  th )
 
Theoremad4ant124 1205 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ( ( ph  /\ 
 ps )  /\  ta )  /\  ch )  ->  th )
 
Theoremad4ant134 1206 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ( ( ph  /\ 
 ta )  /\  ps )  /\  ch )  ->  th )
 
Theoremad4ant234 1207 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ( ( ta 
 /\  ph )  /\  ps )  /\  ch )  ->  th )
 
Theorem3an1rs 1208 Swap conjuncts. (Contributed by NM, 16-Dec-2007.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ta )   =>    |-  (
 ( ( ph  /\  ps  /\ 
 th )  /\  ch )  ->  ta )
 
Theorem3imp1 1209 Importation to left triple conjunction. (Contributed by NM, 24-Feb-2005.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( (
 ph  /\  ps  /\  ch )  /\  th )  ->  ta )
 
Theorem3impd 1210 Importation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch  /\ 
 th )  ->  ta )
 )
 
Theorem3imp2 1211 Importation to right triple conjunction. (Contributed by NM, 26-Oct-2006.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( ph  /\  ( ps  /\  ch  /\ 
 th ) )  ->  ta )
 
Theorem3exp1 1212 Exportation from left triple conjunction. (Contributed by NM, 24-Feb-2005.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) ) )
 
Theorem3expd 1213 Exportation deduction for triple conjunction. (Contributed by NM, 26-Oct-2006.)
 |-  ( ph  ->  (
 ( ps  /\  ch  /\ 
 th )  ->  ta )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theorem3exp2 1214 Exportation from right triple conjunction. (Contributed by NM, 26-Oct-2006.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta ) ) ) )
 
Theoremexp5o 1215 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  (
 ( th  /\  ta )  ->  et ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp516 1216 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
 |-  ( ( ( ph  /\  ( ps  /\  ch  /\ 
 th ) )  /\  ta )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp520 1217 A triple exportation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  ( th  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theorem3anassrs 1218 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  ta )   =>    |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  ->  ta )
 
Theorem3adant1l 1219 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ( ta  /\  ph )  /\  ps  /\  ch )  ->  th )
 
Theorem3adant1r 1220 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ( ph  /\  ta )  /\  ps  /\  ch )  ->  th )
 
Theorem3adant2l 1221 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ( ta 
 /\  ps )  /\  ch )  ->  th )
 
Theorem3adant2r 1222 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ( ps 
 /\  ta )  /\  ch )  ->  th )
 
Theorem3adant3l 1223 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ps  /\  ( ta  /\  ch )
 )  ->  th )
 
Theorem3adant3r 1224 Deduction adding a conjunct to antecedent. (Contributed by NM, 8-Jan-2006.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ps  /\  ( ch  /\  ta )
 )  ->  th )
 
Theoremsyl12anc 1225 Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ( ps 
 /\  ( ch  /\  th ) )  ->  ta )   =>    |-  ( ph  ->  ta )
 
Theoremsyl21anc 1226 Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ( ( ps  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ph  ->  ta )
 
Theoremsyl3anc 1227 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ph  ->  ta )
 
Theoremsyl22anc 1228 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  et )
 
Theoremsyl13anc 1229 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ( ps 
 /\  ( ch  /\  th 
 /\  ta ) )  ->  et )   =>    |-  ( ph  ->  et )
 
Theoremsyl31anc 1230 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ta )  ->  et )   =>    |-  ( ph  ->  et )
 
Theoremsyl112anc 1231 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ( ps 
 /\  ch  /\  ( th  /\ 
 ta ) )  ->  et )   =>    |-  ( ph  ->  et )
 
Theoremsyl121anc 1232 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ( ps 
 /\  ( ch  /\  th )  /\  ta )  ->  et )   =>    |-  ( ph  ->  et )
 
Theoremsyl211anc 1233 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ( ( ps  /\  ch )  /\  th  /\  ta )  ->  et )   =>    |-  ( ph  ->  et )
 
Theoremsyl23anc 1234 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta  /\ 
 et ) )  ->  ze )   =>    |-  ( ph  ->  ze )
 
Theoremsyl32anc 1235 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ( ta 
 /\  et ) )  ->  ze )   =>    |-  ( ph  ->  ze )
 
Theoremsyl122anc 1236 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ( ps 
 /\  ( ch  /\  th )  /\  ( ta 
 /\  et ) )  ->  ze )   =>    |-  ( ph  ->  ze )
 
Theoremsyl212anc 1237 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ( ( ps  /\  ch )  /\  th  /\  ( ta 
 /\  et ) )  ->  ze )   =>    |-  ( ph  ->  ze )
 
Theoremsyl221anc 1238 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta )  /\  et )  ->  ze )   =>    |-  ( ph  ->  ze )
 
Theoremsyl113anc 1239 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ( ps 
 /\  ch  /\  ( th  /\ 
 ta  /\  et )
 )  ->  ze )   =>    |-  ( ph  ->  ze )
 
Theoremsyl131anc 1240 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ( ps 
 /\  ( ch  /\  th 
 /\  ta )  /\  et )  ->  ze )   =>    |-  ( ph  ->  ze )
 
Theoremsyl311anc 1241 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ta  /\  et )  ->  ze )   =>    |-  ( ph  ->  ze )
 
Theoremsyl33anc 1242 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ( ta 
 /\  et  /\  ze )
 )  ->  si )   =>    |-  ( ph  ->  si )
 
Theoremsyl222anc 1243 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta )  /\  ( et  /\  ze ) )  ->  si )   =>    |-  ( ph  ->  si )
 
Theoremsyl123anc 1244 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ( ps 
 /\  ( ch  /\  th )  /\  ( ta 
 /\  et  /\  ze )
 )  ->  si )   =>    |-  ( ph  ->  si )
 
Theoremsyl132anc 1245 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ( ps 
 /\  ( ch  /\  th 
 /\  ta )  /\  ( et  /\  ze ) ) 
 ->  si )   =>    |-  ( ph  ->  si )
 
Theoremsyl213anc 1246 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ( ( ps  /\  ch )  /\  th  /\  ( ta 
 /\  et  /\  ze )
 )  ->  si )   =>    |-  ( ph  ->  si )
 
Theoremsyl231anc 1247 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta  /\ 
 et )  /\  ze )  ->  si )   =>    |-  ( ph  ->  si )
 
Theoremsyl312anc 1248 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ta  /\  ( et  /\  ze )
 )  ->  si )   =>    |-  ( ph  ->  si )
 
Theoremsyl321anc 1249 Syllogism combined with contraction. (Contributed by NM, 11-Jul-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ( ta 
 /\  et )  /\  ze )  ->  si )   =>    |-  ( ph  ->  si )
 
Theoremsyl133anc 1250 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ph  ->  si )   &    |-  ( ( ps 
 /\  ( ch  /\  th 
 /\  ta )  /\  ( et  /\  ze  /\  si ) )  ->  rh )   =>    |-  ( ph  ->  rh )
 
Theoremsyl313anc 1251 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ph  ->  si )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ta  /\  ( et  /\  ze  /\  si ) )  ->  rh )   =>    |-  ( ph  ->  rh )
 
Theoremsyl331anc 1252 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ph  ->  si )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ( ta 
 /\  et  /\  ze )  /\  si )  ->  rh )   =>    |-  ( ph  ->  rh )
 
Theoremsyl223anc 1253 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ph  ->  si )   &    |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta )  /\  ( et  /\  ze 
 /\  si ) )  ->  rh )   =>    |-  ( ph  ->  rh )
 
Theoremsyl232anc 1254 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ph  ->  si )   &    |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta  /\ 
 et )  /\  ( ze  /\  si ) ) 
 ->  rh )   =>    |-  ( ph  ->  rh )
 
Theoremsyl322anc 1255 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ph  ->  si )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ( ta 
 /\  et )  /\  ( ze  /\  si ) ) 
 ->  rh )   =>    |-  ( ph  ->  rh )
 
Theoremsyl233anc 1256 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ph  ->  si )   &    |-  ( ph  ->  rh )   &    |-  ( ( ( ps  /\  ch )  /\  ( th  /\  ta  /\ 
 et )  /\  ( ze  /\  si  /\  rh )
 )  ->  mu )   =>    |-  ( ph  ->  mu )
 
Theoremsyl323anc 1257 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ph  ->  si )   &    |-  ( ph  ->  rh )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ( ta 
 /\  et )  /\  ( ze  /\  si  /\  rh )
 )  ->  mu )   =>    |-  ( ph  ->  mu )
 
Theoremsyl332anc 1258 Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ph  ->  si )   &    |-  ( ph  ->  rh )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ( ta 
 /\  et  /\  ze )  /\  ( si  /\  rh ) )  ->  mu )   =>    |-  ( ph  ->  mu )
 
Theoremsyl333anc 1259 A syllogism inference combined with contraction. (Contributed by NM, 10-Mar-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  et )   &    |-  ( ph  ->  ze )   &    |-  ( ph  ->  si )   &    |-  ( ph  ->  rh )   &    |-  ( ph  ->  mu )   &    |-  ( ( ( ps  /\  ch  /\  th )  /\  ( ta 
 /\  et  /\  ze )  /\  ( si  /\  rh  /\ 
 mu ) )  ->  la )   =>    |-  ( ph  ->  la )
 
Theoremsyl3an1 1260 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ph  ->  ps )   &    |-  (
 ( ps  /\  ch  /\ 
 th )  ->  ta )   =>    |-  (
 ( ph  /\  ch  /\  th )  ->  ta )
 
Theoremsyl3an2 1261 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ph  ->  ch )   &    |-  (
 ( ps  /\  ch  /\ 
 th )  ->  ta )   =>    |-  (
 ( ps  /\  ph  /\  th )  ->  ta )
 
Theoremsyl3an3 1262 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ph  ->  th )   &    |-  (
 ( ps  /\  ch  /\ 
 th )  ->  ta )   =>    |-  (
 ( ps  /\  ch  /\  ph )  ->  ta )
 
Theoremsyl3an1b 1263 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ph  <->  ps )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ( ph  /\  ch  /\ 
 th )  ->  ta )
 
Theoremsyl3an2b 1264 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ph  <->  ch )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ( ps  /\  ph 
 /\  th )  ->  ta )
 
Theoremsyl3an3b 1265 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ph  <->  th )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ( ps  /\  ch 
 /\  ph )  ->  ta )
 
Theoremsyl3an1br 1266 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ps  <->  ph )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ( ph  /\  ch  /\ 
 th )  ->  ta )
 
Theoremsyl3an2br 1267 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ch  <->  ph )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ( ps  /\  ph 
 /\  th )  ->  ta )
 
Theoremsyl3an3br 1268 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( th  <->  ph )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ( ps  /\  ch 
 /\  ph )  ->  ta )
 
Theoremsyl3an 1269 A triple syllogism inference. (Contributed by NM, 13-May-2004.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  th )   &    |-  ( ta  ->  et )   &    |-  ( ( ps 
 /\  th  /\  et )  ->  ze )   =>    |-  ( ( ph  /\  ch  /\ 
 ta )  ->  ze )
 
Theoremsyl3anb 1270 A triple syllogism inference. (Contributed by NM, 15-Oct-2005.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   &    |-  ( ta  <->  et )   &    |-  ( ( ps 
 /\  th  /\  et )  ->  ze )   =>    |-  ( ( ph  /\  ch  /\ 
 ta )  ->  ze )
 
Theoremsyl3anbr 1271 A triple syllogism inference. (Contributed by NM, 29-Dec-2011.)
 |-  ( ps  <->  ph )   &    |-  ( th  <->  ch )   &    |-  ( et  <->  ta )   &    |-  ( ( ps 
 /\  th  /\  et )  ->  ze )   =>    |-  ( ( ph  /\  ch  /\ 
 ta )  ->  ze )
 
Theoremsyld3an3 1272 A syllogism inference. (Contributed by NM, 20-May-2007.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   &    |-  (
 ( ph  /\  ps  /\  th )  ->  ta )   =>    |-  (
 ( ph  /\  ps  /\  ch )  ->  ta )
 
Theoremsyld3an1 1273 A syllogism inference. (Contributed by NM, 7-Jul-2008.)
 |-  ( ( ch  /\  ps 
 /\  th )  ->  ph )   &    |-  (
 ( ph  /\  ps  /\  th )  ->  ta )   =>    |-  (
 ( ch  /\  ps  /\ 
 th )  ->  ta )
 
Theoremsyld3an2 1274 A syllogism inference. (Contributed by NM, 20-May-2007.)
 |-  ( ( ph  /\  ch  /\ 
 th )  ->  ps )   &    |-  (
 ( ph  /\  ps  /\  th )  ->  ta )   =>    |-  (
 ( ph  /\  ch  /\  th )  ->  ta )
 
Theoremsyl3anl1 1275 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
 |-  ( ph  ->  ps )   &    |-  (
 ( ( ps  /\  ch 
 /\  th )  /\  ta )  ->  et )   =>    |-  ( ( (
 ph  /\  ch  /\  th )  /\  ta )  ->  et )
 
Theoremsyl3anl2 1276 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
 |-  ( ph  ->  ch )   &    |-  (
 ( ( ps  /\  ch 
 /\  th )  /\  ta )  ->  et )   =>    |-  ( ( ( ps  /\  ph  /\  th )  /\  ta )  ->  et )
 
Theoremsyl3anl3 1277 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
 |-  ( ph  ->  th )   &    |-  (
 ( ( ps  /\  ch 
 /\  th )  /\  ta )  ->  et )   =>    |-  ( ( ( ps  /\  ch  /\  ph )  /\  ta )  ->  et )
 
Theoremsyl3anl 1278 A triple syllogism inference. (Contributed by NM, 24-Dec-2006.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  th )   &    |-  ( ta  ->  et )   &    |-  ( ( ( ps  /\  th  /\  et )  /\  ze )  ->  si )   =>    |-  ( ( ( ph  /\ 
 ch  /\  ta )  /\  ze )  ->  si )
 
Theoremsyl3anr1 1279 A syllogism inference. (Contributed by NM, 31-Jul-2007.)
 |-  ( ph  ->  ps )   &    |-  (
 ( ch  /\  ( ps  /\  th  /\  ta ) )  ->  et )   =>    |-  (
 ( ch  /\  ( ph  /\  th  /\  ta ) )  ->  et )
 
Theoremsyl3anr2 1280 A syllogism inference. (Contributed by NM, 1-Aug-2007.)
 |-  ( ph  ->  th )   &    |-  (
 ( ch  /\  ( ps  /\  th  /\  ta ) )  ->  et )   =>    |-  (
 ( ch  /\  ( ps  /\  ph  /\  ta )
 )  ->  et )
 
Theoremsyl3anr3 1281 A syllogism inference. (Contributed by NM, 23-Aug-2007.)
 |-  ( ph  ->  ta )   &    |-  (
 ( ch  /\  ( ps  /\  th  /\  ta ) )  ->  et )   =>    |-  (
 ( ch  /\  ( ps  /\  th  /\  ph )
 )  ->  et )
 
Theorem3impdi 1282 Importation inference (undistribute conjunction). (Contributed by NM, 14-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ph  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorem3impdir 1283 Importation inference (undistribute conjunction). (Contributed by NM, 20-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  ps ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ch  /\ 
 ps )  ->  th )
 
Theorem3anidm12 1284 Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
 |-  ( ( ph  /\  ph  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theorem3anidm13 1285 Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
 |-  ( ( ph  /\  ps  /\  ph )  ->  ch )   =>    |-  (
 ( ph  /\  ps )  ->  ch )
 
Theorem3anidm23 1286 Inference from idempotent law for conjunction. (Contributed by NM, 1-Feb-2007.)
 |-  ( ( ph  /\  ps  /\ 
 ps )  ->  ch )   =>    |-  (
 ( ph  /\  ps )  ->  ch )
 
Theoremsyl2an3an 1287 syl3an 1269 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( th  ->  ta )   &    |-  ( ( ps 
 /\  ch  /\  ta )  ->  et )   =>    |-  ( ( ph  /\  th )  ->  et )
 
Theoremsyl2an23an 1288 Deduction related to syl3an 1269 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ( th  /\  ph )  ->  ta )   &    |-  (
 ( ps  /\  ch  /\ 
 ta )  ->  et )   =>    |-  (
 ( th  /\  ph )  ->  et )
 
Theorem3ori 1289 Infer implication from triple disjunction. (Contributed by NM, 26-Sep-2006.)
 |-  ( ph  \/  ps  \/  ch )   =>    |-  ( ( -.  ph  /\ 
 -.  ps )  ->  ch )
 
Theorem3jao 1290 Disjunction of 3 antecedents. (Contributed by NM, 8-Apr-1994.)
 |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  ps )  /\  ( th  ->  ps ) )  ->  ( ( ph  \/  ch 
 \/  th )  ->  ps )
 )
 
Theorem3jaob 1291 Disjunction of 3 antecedents. (Contributed by NM, 13-Sep-2011.)
 |-  ( ( ( ph  \/  ch  \/  th )  ->  ps )  <->  ( ( ph  ->  ps )  /\  ( ch  ->  ps )  /\  ( th  ->  ps ) ) )
 
Theorem3jaoi 1292 Disjunction of 3 antecedents (inference). (Contributed by NM, 12-Sep-1995.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  ps )   &    |-  ( th  ->  ps )   =>    |-  ( ( ph  \/  ch 
 \/  th )  ->  ps )
 
Theorem3jaod 1293 Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  ->  ch ) )   &    |-  ( ph  ->  ( ta  ->  ch )
 )   =>    |-  ( ph  ->  (
 ( ps  \/  th  \/  ta )  ->  ch )
 )
 
Theorem3jaoian 1294 Disjunction of 3 antecedents (inference). (Contributed by NM, 14-Oct-2005.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( th  /\ 
 ps )  ->  ch )   &    |-  (
 ( ta  /\  ps )  ->  ch )   =>    |-  ( ( ( ph  \/  th  \/  ta )  /\  ps )  ->  ch )
 
Theorem3jaodan 1295 Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 th )  ->  ch )   &    |-  (
 ( ph  /\  ta )  ->  ch )   =>    |-  ( ( ph  /\  ( ps  \/  th  \/  ta ) )  ->  ch )
 
Theoremmpjao3dan 1296 Eliminate a 3-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 th )  ->  ch )   &    |-  (
 ( ph  /\  ta )  ->  ch )   &    |-  ( ph  ->  ( ps  \/  th  \/  ta ) )   =>    |-  ( ph  ->  ch )
 
Theorem3jaao 1297 Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ta  ->  ch ) )   &    |-  ( et  ->  ( ze  ->  ch )
 )   =>    |-  ( ( ph  /\  th  /\ 
 et )  ->  (
 ( ps  \/  ta  \/  ze )  ->  ch )
 )
 
Theorem3ianorr 1298 Triple disjunction implies negated triple conjunction. (Contributed by Jim Kingdon, 23-Dec-2018.)
 |-  ( ( -.  ph  \/  -.  ps  \/  -.  ch )  ->  -.  ( ph  /\  ps  /\  ch ) )
 
Theoremsyl3an9b 1299 Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ch  <->  ta ) )   &    |-  ( et  ->  ( ta  <->  ze ) )   =>    |-  ( ( ph  /\ 
 th  /\  et )  ->  ( ps  <->  ze ) )
 
Theorem3orbi123d 1300 Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   &    |-  ( ph  ->  ( et  <->  ze ) )   =>    |-  ( ph  ->  ( ( ps  \/  th  \/  et )  <->  ( ch  \/  ta 
 \/  ze ) ) )
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