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Theorem List for Metamath Proof Explorer - 1101-1200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsimp1r1 1101 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\  ( ph  /\  ps  /\ 
 ch ) )  /\  ta 
 /\  et )  ->  ph )
 
Theoremsimp1r2 1102 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\  ( ph  /\  ps  /\ 
 ch ) )  /\  ta 
 /\  et )  ->  ps )
 
Theoremsimp1r3 1103 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\  ( ph  /\  ps  /\ 
 ch ) )  /\  ta 
 /\  et )  ->  ch )
 
Theoremsimp2l1 1104 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ( ph  /\  ps  /\ 
 ch )  /\  th )  /\  et )  ->  ph )
 
Theoremsimp2l2 1105 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ( ph  /\  ps  /\ 
 ch )  /\  th )  /\  et )  ->  ps )
 
Theoremsimp2l3 1106 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ( ph  /\  ps  /\ 
 ch )  /\  th )  /\  et )  ->  ch )
 
Theoremsimp2r1 1107 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )
 )  /\  et )  -> 
 ph )
 
Theoremsimp2r2 1108 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )
 )  /\  et )  ->  ps )
 
Theoremsimp2r3 1109 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )
 )  /\  et )  ->  ch )
 
Theoremsimp3l1 1110 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( ( ph  /\ 
 ps  /\  ch )  /\  th ) )  ->  ph )
 
Theoremsimp3l2 1111 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( ( ph  /\ 
 ps  /\  ch )  /\  th ) )  ->  ps )
 
Theoremsimp3l3 1112 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( ( ph  /\ 
 ps  /\  ch )  /\  th ) )  ->  ch )
 
Theoremsimp3r1 1113 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( th  /\  ( ph  /\  ps  /\  ch ) ) )  ->  ph )
 
Theoremsimp3r2 1114 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( th  /\  ( ph  /\  ps  /\  ch ) ) )  ->  ps )
 
Theoremsimp3r3 1115 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( th  /\  ( ph  /\  ps  /\  ch ) ) )  ->  ch )
 
Theoremsimp11l 1116 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps )  /\  ch  /\  th )  /\  ta  /\  et )  -> 
 ph )
 
Theoremsimp11r 1117 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps )  /\  ch  /\  th )  /\  ta  /\  et )  ->  ps )
 
Theoremsimp12l 1118 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( ch 
 /\  ( ph  /\  ps )  /\  th )  /\  ta 
 /\  et )  ->  ph )
 
Theoremsimp12r 1119 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( ch 
 /\  ( ph  /\  ps )  /\  th )  /\  ta 
 /\  et )  ->  ps )
 
Theoremsimp13l 1120 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( ch 
 /\  th  /\  ( ph  /\ 
 ps ) )  /\  ta 
 /\  et )  ->  ph )
 
Theoremsimp13r 1121 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( ch 
 /\  th  /\  ( ph  /\ 
 ps ) )  /\  ta 
 /\  et )  ->  ps )
 
Theoremsimp21l 1122 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ( ph  /\  ps )  /\  ch  /\  th )  /\  et )  ->  ph )
 
Theoremsimp21r 1123 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ( ph  /\  ps )  /\  ch  /\  th )  /\  et )  ->  ps )
 
Theoremsimp22l 1124 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ch  /\  ( ph  /\ 
 ps )  /\  th )  /\  et )  ->  ph )
 
Theoremsimp22r 1125 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ch  /\  ( ph  /\ 
 ps )  /\  th )  /\  et )  ->  ps )
 
Theoremsimp23l 1126 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ch  /\  th  /\  ( ph  /\  ps )
 )  /\  et )  -> 
 ph )
 
Theoremsimp23r 1127 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  ( ch  /\  th  /\  ( ph  /\  ps )
 )  /\  et )  ->  ps )
 
Theoremsimp31l 1128 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( ( ph  /\ 
 ps )  /\  ch  /\ 
 th ) )  ->  ph )
 
Theoremsimp31r 1129 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( ( ph  /\ 
 ps )  /\  ch  /\ 
 th ) )  ->  ps )
 
Theoremsimp32l 1130 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( ch  /\  ( ph  /\  ps )  /\  th ) )  ->  ph )
 
Theoremsimp32r 1131 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( ch  /\  ( ph  /\  ps )  /\  th ) )  ->  ps )
 
Theoremsimp33l 1132 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( ch  /\  th 
 /\  ( ph  /\  ps ) ) )  ->  ph )
 
Theoremsimp33r 1133 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ta  /\  et  /\  ( ch  /\  th 
 /\  ( ph  /\  ps ) ) )  ->  ps )
 
Theoremsimp111 1134 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps  /\  ch )  /\  th  /\  ta )  /\  et  /\  ze )  ->  ph )
 
Theoremsimp112 1135 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps  /\  ch )  /\  th  /\  ta )  /\  et  /\  ze )  ->  ps )
 
Theoremsimp113 1136 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( (
 ph  /\  ps  /\  ch )  /\  th  /\  ta )  /\  et  /\  ze )  ->  ch )
 
Theoremsimp121 1137 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\  ( ph  /\  ps  /\ 
 ch )  /\  ta )  /\  et  /\  ze )  ->  ph )
 
Theoremsimp122 1138 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\  ( ph  /\  ps  /\ 
 ch )  /\  ta )  /\  et  /\  ze )  ->  ps )
 
Theoremsimp123 1139 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\  ( ph  /\  ps  /\ 
 ch )  /\  ta )  /\  et  /\  ze )  ->  ch )
 
Theoremsimp131 1140 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\ 
 ta  /\  ( ph  /\ 
 ps  /\  ch )
 )  /\  et  /\  ze )  ->  ph )
 
Theoremsimp132 1141 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\ 
 ta  /\  ( ph  /\ 
 ps  /\  ch )
 )  /\  et  /\  ze )  ->  ps )
 
Theoremsimp133 1142 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( ( th  /\ 
 ta  /\  ( ph  /\ 
 ps  /\  ch )
 )  /\  et  /\  ze )  ->  ch )
 
Theoremsimp211 1143 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( ( ph  /\  ps  /\ 
 ch )  /\  th  /\ 
 ta )  /\  ze )  ->  ph )
 
Theoremsimp212 1144 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( ( ph  /\  ps  /\ 
 ch )  /\  th  /\ 
 ta )  /\  ze )  ->  ps )
 
Theoremsimp213 1145 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( ( ph  /\  ps  /\ 
 ch )  /\  th  /\ 
 ta )  /\  ze )  ->  ch )
 
Theoremsimp221 1146 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )  /\  ta )  /\  ze )  ->  ph )
 
Theoremsimp222 1147 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )  /\  ta )  /\  ze )  ->  ps )
 
Theoremsimp223 1148 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ( ph  /\ 
 ps  /\  ch )  /\  ta )  /\  ze )  ->  ch )
 
Theoremsimp231 1149 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ta  /\  ( ph  /\  ps  /\  ch ) )  /\  ze )  ->  ph )
 
Theoremsimp232 1150 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ta  /\  ( ph  /\  ps  /\  ch ) )  /\  ze )  ->  ps )
 
Theoremsimp233 1151 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ( th  /\  ta  /\  ( ph  /\  ps  /\  ch ) )  /\  ze )  ->  ch )
 
Theoremsimp311 1152 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( ( ph  /\ 
 ps  /\  ch )  /\  th  /\  ta )
 )  ->  ph )
 
Theoremsimp312 1153 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( ( ph  /\ 
 ps  /\  ch )  /\  th  /\  ta )
 )  ->  ps )
 
Theoremsimp313 1154 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( ( ph  /\ 
 ps  /\  ch )  /\  th  /\  ta )
 )  ->  ch )
 
Theoremsimp321 1155 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( th  /\  ( ph  /\  ps  /\  ch )  /\  ta )
 )  ->  ph )
 
Theoremsimp322 1156 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( th  /\  ( ph  /\  ps  /\  ch )  /\  ta )
 )  ->  ps )
 
Theoremsimp323 1157 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( th  /\  ( ph  /\  ps  /\  ch )  /\  ta )
 )  ->  ch )
 
Theoremsimp331 1158 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( th  /\  ta 
 /\  ( ph  /\  ps  /\ 
 ch ) ) ) 
 ->  ph )
 
Theoremsimp332 1159 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( th  /\  ta 
 /\  ( ph  /\  ps  /\ 
 ch ) ) ) 
 ->  ps )
 
Theoremsimp333 1160 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
 |-  ( ( et  /\  ze 
 /\  ( th  /\  ta 
 /\  ( ph  /\  ps  /\ 
 ch ) ) ) 
 ->  ch )
 
Theorem3adantl1 1161 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ta 
 /\  ph  /\  ps )  /\  ch )  ->  th )
 
Theorem3adantl2 1162 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ph  /\ 
 ta  /\  ps )  /\  ch )  ->  th )
 
Theorem3adantl3 1163 Deduction adding a conjunct to antecedent. (Contributed by NM, 24-Feb-2005.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ph  /\ 
 ps  /\  ta )  /\  ch )  ->  th )
 
Theorem3adantr1 1164 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ( ta  /\  ps  /\  ch ) )  ->  th )
 
Theorem3adantr2 1165 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ( ps  /\  ta  /\  ch ) )  ->  th )
 
Theorem3adantr3 1166 Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ( ps  /\  ch  /\  ta ) )  ->  th )
 
Theorem3ad2antl1 1167 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
 |-  ( ( ph  /\  ch )  ->  th )   =>    |-  ( ( ( ph  /\ 
 ps  /\  ta )  /\  ch )  ->  th )
 
Theorem3ad2antl2 1168 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
 |-  ( ( ph  /\  ch )  ->  th )   =>    |-  ( ( ( ps 
 /\  ph  /\  ta )  /\  ch )  ->  th )
 
Theorem3ad2antl3 1169 Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)
 |-  ( ( ph  /\  ch )  ->  th )   =>    |-  ( ( ( ps 
 /\  ta  /\  ph )  /\  ch )  ->  th )
 
Theorem3ad2antr1 1170 Deduction adding conjuncts to antecedent. (Contributed by NM, 25-Dec-2007.)
 |-  ( ( ph  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ( ch  /\  ps  /\  ta ) )  ->  th )
 
Theorem3ad2antr2 1171 Deduction adding conjuncts to antecedent. (Contributed by NM, 27-Dec-2007.)
 |-  ( ( ph  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ( ps  /\  ch  /\  ta ) )  ->  th )
 
Theorem3ad2antr3 1172 Deduction adding conjuncts to antecedent. (Contributed by NM, 30-Dec-2007.)
 |-  ( ( ph  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ( ps  /\  ta  /\  ch ) )  ->  th )
 
Theorem3anibar 1173 Remove a hypothesis from the second member of a biimplication. (Contributed by FL, 22-Jul-2008.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ( th 
 <->  ( ch  /\  ta ) ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  ( th  <->  ta ) )
 
Theorem3mix1 1174 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
 |-  ( ph  ->  ( ph  \/  ps  \/  ch ) )
 
Theorem3mix2 1175 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
 |-  ( ph  ->  ( ps  \/  ph  \/  ch )
 )
 
Theorem3mix3 1176 Introduction in triple disjunction. (Contributed by NM, 4-Apr-1995.)
 |-  ( ph  ->  ( ps  \/  ch  \/  ph ) )
 
Theorem3mix1i 1177 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
 |-  ph   =>    |-  ( ph  \/  ps  \/  ch )
 
Theorem3mix2i 1178 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
 |-  ph   =>    |-  ( ps  \/  ph  \/  ch )
 
Theorem3mix3i 1179 Introduction in triple disjunction. (Contributed by Mario Carneiro, 6-Oct-2014.)
 |-  ph   =>    |-  ( ps  \/  ch  \/  ph )
 
Theorem3mix1d 1180 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ps  \/  ch 
 \/  th ) )
 
Theorem3mix2d 1181 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  \/  ps 
 \/  th ) )
 
Theorem3mix3d 1182 Deduction introducing triple disjunction. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  \/  th 
 \/  ps ) )
 
Theorem3pm3.2i 1183 Infer conjunction of premises. (Contributed by NM, 10-Feb-1995.)
 |-  ph   &    |- 
 ps   &    |- 
 ch   =>    |-  ( ph  /\  ps  /\ 
 ch )
 
Theorempm3.2an3 1184 Version of pm3.2 448 for a triple conjunction. (Contributed by Alan Sare, 24-Oct-2011.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  (
 ph  /\  ps  /\  ch ) ) ) )
 
Theorem3jca 1185 Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   =>    |-  ( ph  ->  ( ps  /\  ch  /\  th ) )
 
Theorem3jcad 1186 Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  th ) )   &    |-  ( ph  ->  ( ps  ->  ta )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  th 
 /\  ta ) ) )
 
Theoremmpbir3an 1187 Detach a conjunction of truths in a biconditional. (Contributed by NM, 16-Sep-2011.)
 |- 
 ps   &    |- 
 ch   &    |- 
 th   &    |-  ( ph  <->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ph
 
Theoremmpbir3and 1188 Detach a conjunction of truths in a biconditional. (Contributed by Mario Carneiro, 11-May-2014.) (Revised by Mario Carneiro, 9-Jan-2015.)
 |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ta )   &    |-  ( ph  ->  ( ps  <->  ( ch  /\  th 
 /\  ta ) ) )   =>    |-  ( ph  ->  ps )
 
Theoremsyl3anbrc 1189 Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ta  <->  ( ps  /\  ch 
 /\  th ) )   =>    |-  ( ph  ->  ta )
 
Theorem3anim123i 1190 Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  th )   &    |-  ( ta  ->  et )   =>    |-  ( ( ph  /\  ch  /\ 
 ta )  ->  ( ps  /\  th  /\  et ) )
 
Theorem3anim1i 1191 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 16-Aug-2009.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ph  /\  ch  /\  th )  ->  ( ps  /\ 
 ch  /\  th )
 )
 
Theorem3anim2i 1192 Add two conjuncts to antecedent and consequent. (Contributed by AV, 21-Nov-2019.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ch  /\  ph  /\  th )  ->  ( ch  /\  ps 
 /\  th ) )
 
Theorem3anim3i 1193 Add two conjuncts to antecedent and consequent. (Contributed by Jeff Hankins, 19-Aug-2009.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ch  /\  th  /\  ph )  ->  ( ch 
 /\  th  /\  ps )
 )
 
Theorem3anbi123i 1194 Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   &    |-  ( ta  <->  et )   =>    |-  ( ( ph  /\  ch  /\ 
 ta )  <->  ( ps  /\  th 
 /\  et ) )
 
Theorem3orbi123i 1195 Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   &    |-  ( ta  <->  et )   =>    |-  ( ( ph  \/  ch 
 \/  ta )  <->  ( ps  \/  th 
 \/  et ) )
 
Theorem3anbi1i 1196 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ph  /\  ch  /\ 
 th )  <->  ( ps  /\  ch 
 /\  th ) )
 
Theorem3anbi2i 1197 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ch  /\  ph 
 /\  th )  <->  ( ch  /\  ps 
 /\  th ) )
 
Theorem3anbi3i 1198 Inference adding two conjuncts to each side of a biconditional. (Contributed by NM, 8-Sep-2006.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ch  /\  th 
 /\  ph )  <->  ( ch  /\  th 
 /\  ps ) )
 
Theorem3imp 1199 Importation inference. (Contributed by NM, 8-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorem3impa 1200 Importation from double to triple conjunction. (Contributed by NM, 20-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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