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Theorem List for Intuitionistic Logic Explorer - 1201-1300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsyl232anc 1201 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 1202 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 1203 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 1204 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 1205 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 1206 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 1207 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ph  ->  ps )   &    |-  (
 ( ps  /\  ch  /\ 
 th )  ->  ta )   =>    |-  (
 ( ph  /\  ch  /\  th )  ->  ta )
 
Theoremsyl3an2 1208 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ph  ->  ch )   &    |-  (
 ( ps  /\  ch  /\ 
 th )  ->  ta )   =>    |-  (
 ( ps  /\  ph  /\  th )  ->  ta )
 
Theoremsyl3an3 1209 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ph  ->  th )   &    |-  (
 ( ps  /\  ch  /\ 
 th )  ->  ta )   =>    |-  (
 ( ps  /\  ch  /\  ph )  ->  ta )
 
Theoremsyl3an1b 1210 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ph  <->  ps )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ( ph  /\  ch  /\ 
 th )  ->  ta )
 
Theoremsyl3an2b 1211 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ph  <->  ch )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ( ps  /\  ph 
 /\  th )  ->  ta )
 
Theoremsyl3an3b 1212 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ph  <->  th )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ( ps  /\  ch 
 /\  ph )  ->  ta )
 
Theoremsyl3an1br 1213 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ps  <->  ph )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ( ph  /\  ch  /\ 
 th )  ->  ta )
 
Theoremsyl3an2br 1214 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( ch  <->  ph )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ( ps  /\  ph 
 /\  th )  ->  ta )
 
Theoremsyl3an3br 1215 A syllogism inference. (Contributed by NM, 22-Aug-1995.)
 |-  ( th  <->  ph )   &    |-  ( ( ps 
 /\  ch  /\  th )  ->  ta )   =>    |-  ( ( ps  /\  ch 
 /\  ph )  ->  ta )
 
Theoremsyl3an 1216 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 1217 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 1218 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 1219 A syllogism inference. (Contributed by NM, 20-May-2007.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   &    |-  (
 ( ph  /\  ps  /\  th )  ->  ta )   =>    |-  (
 ( ph  /\  ps  /\  ch )  ->  ta )
 
Theoremsyld3an1 1220 A syllogism inference. (Contributed by NM, 7-Jul-2008.)
 |-  ( ( ch  /\  ps 
 /\  th )  ->  ph )   &    |-  (
 ( ph  /\  ps  /\  th )  ->  ta )   =>    |-  (
 ( ch  /\  ps  /\ 
 th )  ->  ta )
 
Theoremsyld3an2 1221 A syllogism inference. (Contributed by NM, 20-May-2007.)
 |-  ( ( ph  /\  ch  /\ 
 th )  ->  ps )   &    |-  (
 ( ph  /\  ps  /\  th )  ->  ta )   =>    |-  (
 ( ph  /\  ch  /\  th )  ->  ta )
 
Theoremsyl3anl1 1222 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
 |-  ( ph  ->  ps )   &    |-  (
 ( ( ps  /\  ch 
 /\  th )  /\  ta )  ->  et )   =>    |-  ( ( (
 ph  /\  ch  /\  th )  /\  ta )  ->  et )
 
Theoremsyl3anl2 1223 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
 |-  ( ph  ->  ch )   &    |-  (
 ( ( ps  /\  ch 
 /\  th )  /\  ta )  ->  et )   =>    |-  ( ( ( ps  /\  ph  /\  th )  /\  ta )  ->  et )
 
Theoremsyl3anl3 1224 A syllogism inference. (Contributed by NM, 24-Feb-2005.)
 |-  ( ph  ->  th )   &    |-  (
 ( ( ps  /\  ch 
 /\  th )  /\  ta )  ->  et )   =>    |-  ( ( ( ps  /\  ch  /\  ph )  /\  ta )  ->  et )
 
Theoremsyl3anl 1225 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 1226 A syllogism inference. (Contributed by NM, 31-Jul-2007.)
 |-  ( ph  ->  ps )   &    |-  (
 ( ch  /\  ( ps  /\  th  /\  ta ) )  ->  et )   =>    |-  (
 ( ch  /\  ( ph  /\  th  /\  ta ) )  ->  et )
 
Theoremsyl3anr2 1227 A syllogism inference. (Contributed by NM, 1-Aug-2007.)
 |-  ( ph  ->  th )   &    |-  (
 ( ch  /\  ( ps  /\  th  /\  ta ) )  ->  et )   =>    |-  (
 ( ch  /\  ( ps  /\  ph  /\  ta )
 )  ->  et )
 
Theoremsyl3anr3 1228 A syllogism inference. (Contributed by NM, 23-Aug-2007.)
 |-  ( ph  ->  ta )   &    |-  (
 ( ch  /\  ( ps  /\  th  /\  ta ) )  ->  et )   =>    |-  (
 ( ch  /\  ( ps  /\  th  /\  ph )
 )  ->  et )
 
Theorem3impdi 1229 Importation inference (undistribute conjunction). (Contributed by NM, 14-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ph  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorem3impdir 1230 Importation inference (undistribute conjunction). (Contributed by NM, 20-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  ps ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ch  /\ 
 ps )  ->  th )
 
Theorem3anidm12 1231 Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
 |-  ( ( ph  /\  ph  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theorem3anidm13 1232 Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
 |-  ( ( ph  /\  ps  /\  ph )  ->  ch )   =>    |-  (
 ( ph  /\  ps )  ->  ch )
 
Theorem3anidm23 1233 Inference from idempotent law for conjunction. (Contributed by NM, 1-Feb-2007.)
 |-  ( ( ph  /\  ps  /\ 
 ps )  ->  ch )   =>    |-  (
 ( ph  /\  ps )  ->  ch )
 
Theoremsyl2an3an 1234 syl3an 1216 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 1235 Deduction related to syl3an 1216 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 1236 Infer implication from triple disjunction. (Contributed by NM, 26-Sep-2006.)
 |-  ( ph  \/  ps  \/  ch )   =>    |-  ( ( -.  ph  /\ 
 -.  ps )  ->  ch )
 
Theorem3jao 1237 Disjunction of 3 antecedents. (Contributed by NM, 8-Apr-1994.)
 |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  ps )  /\  ( th  ->  ps ) )  ->  ( ( ph  \/  ch 
 \/  th )  ->  ps )
 )
 
Theorem3jaob 1238 Disjunction of 3 antecedents. (Contributed by NM, 13-Sep-2011.)
 |-  ( ( ( ph  \/  ch  \/  th )  ->  ps )  <->  ( ( ph  ->  ps )  /\  ( ch  ->  ps )  /\  ( th  ->  ps ) ) )
 
Theorem3jaoi 1239 Disjunction of 3 antecedents (inference). (Contributed by NM, 12-Sep-1995.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  ps )   &    |-  ( th  ->  ps )   =>    |-  ( ( ph  \/  ch 
 \/  th )  ->  ps )
 
Theorem3jaod 1240 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 1241 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 1242 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 1243 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 1244 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 1245 Triple disjunction implies negated triple conjunction. (Contributed by Jim Kingdon, 23-Dec-2018.)
 |-  ( ( -.  ph  \/  -.  ps  \/  -.  ch )  ->  -.  ( ph  /\  ps  /\  ch ) )
 
Theoremsyl3an9b 1246 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 1247 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 ) ) )
 
Theorem3anbi123d 1248 Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   &    |-  ( ph  ->  ( et  <->  ze ) )   =>    |-  ( ph  ->  ( ( ps  /\  th  /\ 
 et )  <->  ( ch  /\  ta 
 /\  ze ) ) )
 
Theorem3anbi12d 1249 Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ( ps  /\  th  /\ 
 et )  <->  ( ch  /\  ta 
 /\  et ) ) )
 
Theorem3anbi13d 1250 Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ( ps  /\  et  /\ 
 th )  <->  ( ch  /\  et  /\  ta ) ) )
 
Theorem3anbi23d 1251 Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ( et  /\  ps  /\ 
 th )  <->  ( et  /\  ch 
 /\  ta ) ) )
 
Theorem3anbi1d 1252 Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( ps  /\  th  /\ 
 ta )  <->  ( ch  /\  th 
 /\  ta ) ) )
 
Theorem3anbi2d 1253 Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( th  /\  ps  /\ 
 ta )  <->  ( th  /\  ch 
 /\  ta ) ) )
 
Theorem3anbi3d 1254 Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( th  /\  ta  /\ 
 ps )  <->  ( th  /\  ta 
 /\  ch ) ) )
 
Theorem3anim123d 1255 Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  ->  ta ) )   &    |-  ( ph  ->  ( et  ->  ze )
 )   =>    |-  ( ph  ->  (
 ( ps  /\  th  /\ 
 et )  ->  ( ch  /\  ta  /\  ze ) ) )
 
Theorem3orim123d 1256 Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  ->  ta ) )   &    |-  ( ph  ->  ( et  ->  ze )
 )   =>    |-  ( ph  ->  (
 ( ps  \/  th  \/  et )  ->  ( ch  \/  ta  \/  ze ) ) )
 
Theoreman6 1257 Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  ( th  /\  ta  /\ 
 et ) )  <->  ( ( ph  /\ 
 th )  /\  ( ps  /\  ta )  /\  ( ch  /\  et )
 ) )
 
Theorem3an6 1258 Analog of an4 553 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th )  /\  ( ta  /\  et )
 ) 
 <->  ( ( ph  /\  ch  /\ 
 ta )  /\  ( ps  /\  th  /\  et ) ) )
 
Theorem3or6 1259 Analog of or4 723 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.)
 |-  ( ( ( ph  \/  ps )  \/  ( ch  \/  th )  \/  ( ta  \/  et ) )  <->  ( ( ph  \/  ch  \/  ta )  \/  ( ps  \/  th  \/  et ) ) )
 
Theoremmp3an1 1260 An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
 |-  ph   &    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ps  /\  ch )  ->  th )
 
Theoremmp3an2 1261 An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
 |- 
 ps   &    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ch )  ->  th )
 
Theoremmp3an3 1262 An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
 |- 
 ch   &    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ps )  ->  th )
 
Theoremmp3an12 1263 An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
 |-  ph   &    |- 
 ps   &    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  ( ch  ->  th )
 
Theoremmp3an13 1264 An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
 |-  ph   &    |- 
 ch   &    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  ( ps  ->  th )
 
Theoremmp3an23 1265 An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
 |- 
 ps   &    |- 
 ch   &    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremmp3an1i 1266 An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.)
 |- 
 ps   &    |-  ( ph  ->  (
 ( ps  /\  ch  /\ 
 th )  ->  ta )
 )   =>    |-  ( ph  ->  (
 ( ch  /\  th )  ->  ta ) )
 
Theoremmp3anl1 1267 An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
 |-  ph   &    |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ta )   =>    |-  (
 ( ( ps  /\  ch )  /\  th )  ->  ta )
 
Theoremmp3anl2 1268 An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
 |- 
 ps   &    |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ta )   =>    |-  (
 ( ( ph  /\  ch )  /\  th )  ->  ta )
 
Theoremmp3anl3 1269 An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
 |- 
 ch   &    |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ta )   =>    |-  (
 ( ( ph  /\  ps )  /\  th )  ->  ta )
 
Theoremmp3anr1 1270 An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.)
 |- 
 ps   &    |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  ta )   =>    |-  (
 ( ph  /\  ( ch 
 /\  th ) )  ->  ta )
 
Theoremmp3anr2 1271 An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.)
 |- 
 ch   &    |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  ta )   =>    |-  (
 ( ph  /\  ( ps 
 /\  th ) )  ->  ta )
 
Theoremmp3anr3 1272 An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.)
 |- 
 th   &    |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  ta )   =>    |-  (
 ( ph  /\  ( ps 
 /\  ch ) )  ->  ta )
 
Theoremmp3an 1273 An inference based on modus ponens. (Contributed by NM, 14-May-1999.)
 |-  ph   &    |- 
 ps   &    |- 
 ch   &    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  th
 
Theoremmpd3an3 1274 An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ps )  ->  th )
 
Theoremmpd3an23 1275 An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremmp3and 1276 A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ( ( ps  /\  ch  /\ 
 th )  ->  ta )
 )   =>    |-  ( ph  ->  ta )
 
Theoremmp3an12i 1277 mp3an 1273 with antecedents in standard conjunction form and with one hypothesis an implication. (Contributed by Alan Sare, 28-Aug-2016.)
 |-  ph   &    |- 
 ps   &    |-  ( ch  ->  th )   &    |-  (
 ( ph  /\  ps  /\  th )  ->  ta )   =>    |-  ( ch  ->  ta )
 
Theoremmp3an2i 1278 mp3an 1273 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  ( ps  ->  th )   &    |-  ( ( ph  /\ 
 ch  /\  th )  ->  ta )   =>    |-  ( ps  ->  ta )
 
Theoremmp3an3an 1279 mp3an 1273 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  ( th  ->  ta )   &    |-  ( ( ph  /\ 
 ch  /\  ta )  ->  et )   =>    |-  ( ( ps  /\  th )  ->  et )
 
Theoremmp3an2ani 1280 An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  (
 ( ps  /\  th )  ->  ta )   &    |-  ( ( ph  /\ 
 ch  /\  ta )  ->  et )   =>    |-  ( ( ps  /\  th )  ->  et )
 
Theorembiimp3a 1281 Infer implication from a logical equivalence. Similar to biimpa 290. (Contributed by NM, 4-Sep-2005.)
 |-  ( ( ph  /\  ps )  ->  ( ch  <->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembiimp3ar 1282 Infer implication from a logical equivalence. Similar to biimpar 291. (Contributed by NM, 2-Jan-2009.)
 |-  ( ( ph  /\  ps )  ->  ( ch  <->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  th )  ->  ch )
 
Theorem3anandis 1283 Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ph  /\  ch )  /\  ( ph  /\  th )
 )  ->  ta )   =>    |-  (
 ( ph  /\  ( ps 
 /\  ch  /\  th )
 )  ->  ta )
 
Theorem3anandirs 1284 Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.) (Revised by NM, 18-Apr-2007.)
 |-  ( ( ( ph  /\ 
 th )  /\  ( ps  /\  th )  /\  ( ch  /\  th )
 )  ->  ta )   =>    |-  (
 ( ( ph  /\  ps  /\ 
 ch )  /\  th )  ->  ta )
 
Theoremecased 1285 Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  ( ps  \/  ch ) )   =>    |-  ( ph  ->  ps )
 
Theoremecase23d 1286 Variation of ecased 1285 with three disjuncts instead of two. (Contributed by NM, 22-Apr-1994.) (Revised by Jim Kingdon, 9-Dec-2017.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  -.  th )   &    |-  ( ph  ->  ( ps  \/  ch  \/  th ) )   =>    |-  ( ph  ->  ps )
 
1.2.13  True and false constants
 
1.2.13.1  Universal quantifier for use by df-tru

Even though it isn't ordinarily part of propositional calculus, the universal quantifier  A. is introduced here so that the soundness of definition df-tru 1292 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in axiom ax-5 1381 in the predicate calculus section below. For those who want propositional calculus to be self-contained i.e. to use wff variables only, the alternate definition dftru2 1297 may be adopted and this subsection moved down to the start of the subsection with wex 1426 below. However, the use of dftru2 1297 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

 
Syntaxwal 1287 Extend wff definition to include the universal quantifier ('for all').  A. x ph is read " ph (phi) is true for all  x." Typically, in its final application 
ph would be replaced with a wff containing a (free) occurrence of the variable  x, for example  x  =  y. In a universe with a finite number of objects, "for all" is equivalent to a big conjunction (AND) with one wff for each possible case of  x. When the universe is infinite (as with set theory), such a propositional-calculus equivalent is not possible because an infinitely long formula has no meaning, but conceptually the idea is the same.
 wff  A. x ph
 
1.2.13.2  Equality predicate for use by df-tru

Even though it isn't ordinarily part of propositional calculus, the equality predicate  = is introduced here so that the soundness of definition df-tru 1292 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in axiom ax-8 1440 in the predicate calculus section below. For those who want propositional calculus to be self-contained i.e. to use wff variables only, the alternate definition dftru2 1297 may be adopted and this subsection moved down to just above weq 1437 below. However, the use of dftru2 1297 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

 
Syntaxcv 1288 This syntax construction states that a variable  x, which has been declared to be a setvar variable by $f statement vx, is also a class expression. This can be justified informally as follows. We know that the class builder  { y  |  y  e.  x } is a class by cab 2074. Since (when  y is distinct from  x) we have  x  =  { y  |  y  e.  x } by cvjust 2083, we can argue that the syntax " class  x " can be viewed as an abbreviation for "
class  { y  |  y  e.  x }". See the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class."

While it is tempting and perhaps occasionally useful to view cv 1288 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1288 is intrinsically no different from any other class-building syntax such as cab 2074, cun 2995, or c0 3284.

For a general discussion of the theory of classes and the role of cv 1288, see http://us.metamath.org/mpeuni/mmset.html#class.

(The description above applies to set theory, not predicate calculus. The purpose of introducing  class  x here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1437 of predicate calculus from the wceq 1289 of set theory, so that we don't overload the  = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers.)

 class  x
 
Syntaxwceq 1289 Extend wff definition to include class equality.

For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class.

(The purpose of introducing 
wff  A  =  B here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1437 of predicate calculus in terms of the wceq 1289 of set theory, so that we don't "overload" the  = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. For example, some parsers - although not the Metamath program - stumble on the fact that the  = in  x  =  y could be the  = of either weq 1437 or wceq 1289, although mathematically it makes no difference. The class variables  A and  B are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-cleq 2081 for more information on the set theory usage of wceq 1289.)

 wff  A  =  B
 
1.2.13.3  Define the true and false constants
 
Syntaxwtru 1290 T. is a wff.
 wff T.
 
Theoremtrujust 1291 Soundness justification theorem for df-tru 1292. (Contributed by Mario Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.)
 |-  ( ( A. x  x  =  x  ->  A. x  x  =  x )  <->  ( A. y  y  =  y  ->  A. y  y  =  y ) )
 
Definitiondf-tru 1292 Definition of the truth value "true", or "verum", denoted by T.. This is a tautology, as proved by tru 1293. In this definition, an instance of id 19 is used as the definiens, although any tautology, such as an axiom, can be used in its place. This particular id 19 instance was chosen so this definition can be checked by the same algorithm that is used for predicate calculus. This definition should be referenced directly only by tru 1293, and other proofs should depend on tru 1293 (directly or indirectly) instead of this definition, since there are many alternate ways to define T.. (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by NM, 11-Jul-2019.) (New usage is discouraged.)
 |-  ( T.  <->  ( A. x  x  =  x  ->  A. x  x  =  x ) )
 
Theoremtru 1293 The truth value T. is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
 |- T.
 
Syntaxwfal 1294 F. is a wff.
 wff F.
 
Definitiondf-fal 1295 Definition of the truth value "false", or "falsum", denoted by F.. See also df-tru 1292. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( F.  <->  -. T.  )
 
Theoremfal 1296 The truth value F. is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)
 |- 
 -. F.
 
Theoremdftru2 1297 An alternate definition of "true". (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) (New usage is discouraged.)
 |-  ( T.  <->  ( ph  ->  ph ) )
 
Theoremmptru 1298 Eliminate T. as an antecedent. A proposition implied by T. is true. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  ( T.  ->  ph )   =>    |-  ph
 
Theoremtbtru 1299 A proposition is equivalent to itself being equivalent to T.. (Contributed by Anthony Hart, 14-Aug-2011.)
 |-  ( ph  <->  ( ph  <-> T.  ) )
 
Theoremnbfal 1300 The negation of a proposition is equivalent to itself being equivalent to F.. (Contributed by Anthony Hart, 14-Aug-2011.)
 |-  ( -.  ph  <->  ( ph  <-> F.  ) )
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