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Theorem List for Intuitionistic Logic Explorer - 201-300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem3imtr4d 201 More general version of 3imtr4i 199. Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  <->  ps ) )   &    |-  ( ph  ->  ( ta  <->  ch ) )   =>    |-  ( ph  ->  ( th  ->  ta )
 )
 
Theorem3imtr3g 202 More general version of 3imtr3i 198. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ps 
 <-> 
 th )   &    |-  ( ch  <->  ta )   =>    |-  ( ph  ->  ( th  ->  ta ) )
 
Theorem3imtr4g 203 More general version of 3imtr4i 199. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th 
 <->  ps )   &    |-  ( ta  <->  ch )   =>    |-  ( ph  ->  ( th  ->  ta ) )
 
Theorem3bitri 204 A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ps  <->  ch )   &    |-  ( ch  <->  th )   =>    |-  ( ph  <->  th )
 
Theorem3bitrri 205 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  <->  ps )   &    |-  ( ps  <->  ch )   &    |-  ( ch  <->  th )   =>    |-  ( th  <->  ph )
 
Theorem3bitr2i 206 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ph  <->  th )
 
Theorem3bitr2ri 207 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( th  <->  ph )
 
Theorem3bitr3i 208 A chained inference from transitive law for logical equivalence. (Contributed by NM, 19-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ph  <->  ch )   &    |-  ( ps  <->  th )   =>    |-  ( ch  <->  th )
 
Theorem3bitr3ri 209 A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ph  <->  ch )   &    |-  ( ps  <->  th )   =>    |-  ( th  <->  ch )
 
Theorem3bitr4i 210 A chained inference from transitive law for logical equivalence. This inference is frequently used to apply a definition to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  ph )   &    |-  ( th  <->  ps )   =>    |-  ( ch  <->  th )
 
Theorem3bitr4ri 211 A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  ph )   &    |-  ( th  <->  ps )   =>    |-  ( th  <->  ch )
 
Theorem3bitrd 212 Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( ch  <->  th ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ps  <->  ta ) )
 
Theorem3bitrrd 213 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( ch  <->  th ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ta  <->  ps ) )
 
Theorem3bitr2d 214 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ps  <->  ta ) )
 
Theorem3bitr2rd 215 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ta  <->  ps ) )
 
Theorem3bitr3d 216 Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 24-Apr-1996.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( ps  <->  th ) )   &    |-  ( ph  ->  ( ch  <->  ta ) )   =>    |-  ( ph  ->  ( th  <->  ta ) )
 
Theorem3bitr3rd 217 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( ps  <->  th ) )   &    |-  ( ph  ->  ( ch  <->  ta ) )   =>    |-  ( ph  ->  ( ta  <->  th ) )
 
Theorem3bitr4d 218 Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 18-Oct-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ps ) )   &    |-  ( ph  ->  ( ta  <->  ch ) )   =>    |-  ( ph  ->  ( th  <->  ta ) )
 
Theorem3bitr4rd 219 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ps ) )   &    |-  ( ph  ->  ( ta  <->  ch ) )   =>    |-  ( ph  ->  ( ta  <->  th ) )
 
Theorem3bitr3g 220 More general version of 3bitr3i 208. Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ps 
 <-> 
 th )   &    |-  ( ch  <->  ta )   =>    |-  ( ph  ->  ( th 
 <->  ta ) )
 
Theorem3bitr4g 221 More general version of 3bitr4i 210. Useful for converting definitions in a formula. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th 
 <->  ps )   &    |-  ( ta  <->  ch )   =>    |-  ( ph  ->  ( th 
 <->  ta ) )
 
Theorembi3ant 222 Construct a biconditional in antecedent position. (Contributed by Wolf Lammen, 14-May-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ( th  ->  ta )  -> 
 ph )  ->  (
 ( ( ta  ->  th )  ->  ps )  ->  ( ( th  <->  ta )  ->  ch )
 ) )
 
Theorembisym 223 Express symmetries of theorems in terms of biconditionals. (Contributed by Wolf Lammen, 14-May-2013.)
 |-  ( ( ( ph  ->  ps )  ->  ( ch  ->  th ) )  ->  ( ( ( ps 
 ->  ph )  ->  ( th  ->  ch ) )  ->  ( ( ph  <->  ps )  ->  ( ch 
 <-> 
 th ) ) ) )
 
Theoremimbi2i 224 Introduce an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 6-Feb-2013.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ch  ->  ph )  <->  ( ch  ->  ps ) )
 
Theorembibi2i 225 Inference adding a biconditional to the left in an equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 16-May-2013.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ch  <->  ph )  <->  ( ch  <->  ps ) )
 
Theorembibi1i 226 Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ph  <->  ch )  <->  ( ps  <->  ch ) )
 
Theorembibi12i 227 The equivalence of two equivalences. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  <->  ch )  <->  ( ps  <->  th ) )
 
Theoremimbi2d 228 Deduction adding an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( th  ->  ps )  <->  ( th  ->  ch )
 ) )
 
Theoremimbi1d 229 Deduction adding a consequent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 17-Sep-2013.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( ps  ->  th )  <->  ( ch  ->  th )
 ) )
 
Theorembibi2d 230 Deduction adding a biconditional to the left in an equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( th  <->  ps )  <->  ( th  <->  ch ) ) )
 
Theorembibi1d 231 Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( ps  <->  th )  <->  ( ch  <->  th ) ) )
 
Theoremimbi12d 232 Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ( ps  ->  th )  <->  ( ch  ->  ta )
 ) )
 
Theorembibi12d 233 Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ( ps  <->  th )  <->  ( ch  <->  ta ) ) )
 
Theoremimbi1 234 Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  <->  ps )  ->  (
 ( ph  ->  ch )  <->  ( ps  ->  ch )
 ) )
 
Theoremimbi2 235 Theorem *4.85 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)
 |-  ( ( ph  <->  ps )  ->  (
 ( ch  ->  ph )  <->  ( ch  ->  ps )
 ) )
 
Theoremimbi1i 236 Introduce a consequent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 17-Sep-2013.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ph  ->  ch )  <->  ( ps  ->  ch ) )
 
Theoremimbi12i 237 Join two logical equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  ->  ch )  <->  ( ps  ->  th ) )
 
Theorembibi1 238 Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  <->  ps )  ->  (
 ( ph  <->  ch )  <->  ( ps  <->  ch ) ) )
 
Theorembiimt 239 A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.)
 |-  ( ph  ->  ( ps 
 <->  ( ph  ->  ps )
 ) )
 
Theorempm5.5 240 Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  ->  (
 ( ph  ->  ps )  <->  ps ) )
 
Theorema1bi 241 Inference introducing a theorem as an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
 |-  ph   =>    |-  ( ps  <->  ( ph  ->  ps ) )
 
Theorempm5.501 242 Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 24-Jan-2013.)
 |-  ( ph  ->  ( ps 
 <->  ( ph  <->  ps ) ) )
 
Theoremibib 243 Implication in terms of implication and biconditional. (Contributed by NM, 31-Mar-1994.) (Proof shortened by Wolf Lammen, 24-Jan-2013.)
 |-  ( ( ph  ->  ps )  <->  ( ph  ->  (
 ph 
 <->  ps ) ) )
 
Theoremibibr 244 Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.)
 |-  ( ( ph  ->  ps )  <->  ( ph  ->  ( ps  <->  ph ) ) )
 
Theoremtbt 245 A wff is equivalent to its equivalence with truth. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ph   =>    |-  ( ps  <->  ( ps  <->  ph ) )
 
Theorembi2.04 246 Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 ) 
 <->  ( ps  ->  ( ph  ->  ch ) ) )
 
Theorempm5.4 247 Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  ->  (
 ph  ->  ps ) )  <->  ( ph  ->  ps ) )
 
Theoremimdi 248 Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 ) 
 <->  ( ( ph  ->  ps )  ->  ( ph  ->  ch ) ) )
 
Theorempm5.41 249 Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 12-Oct-2012.)
 |-  ( ( ( ph  ->  ps )  ->  ( ph  ->  ch ) )  <->  ( ph  ->  ( ps  ->  ch )
 ) )
 
Theoremimim21b 250 Simplify an implication between two implications when the antecedent of the first is a consequence of the antecedent of the second. The reverse form is useful in producing the successor step in induction proofs. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Wolf Lammen, 14-Sep-2013.)
 |-  ( ( ps  ->  ph )  ->  ( (
 ( ph  ->  ch )  ->  ( ps  ->  th )
 ) 
 <->  ( ps  ->  ( ch  ->  th ) ) ) )
 
Theoremimpd 251 Importation deduction. (Contributed by NM, 31-Mar-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch )  ->  th ) )
 
Theoremimp31 252 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ( (
 ph  /\  ps )  /\  ch )  ->  th )
 
Theoremimp32 253 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ( ph  /\  ( ps  /\  ch ) )  ->  th )
 
Theoremexpd 254 Exportation deduction. (Contributed by NM, 20-Aug-1993.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
 
Theoremexpdimp 255 A deduction version of exportation, followed by importation. (Contributed by NM, 6-Sep-2008.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ( ch  ->  th ) )
 
Theoremimpancom 256 Mixed importation/commutation inference. (Contributed by NM, 22-Jun-2013.)
 |-  ( ( ph  /\  ps )  ->  ( ch  ->  th ) )   =>    |-  ( ( ph  /\  ch )  ->  ( ps  ->  th ) )
 
Theorempm3.3 257 Theorem *3.3 (Exp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
 |-  ( ( ( ph  /\ 
 ps )  ->  ch )  ->  ( ph  ->  ( ps  ->  ch ) ) )
 
Theorempm3.31 258 Theorem *3.31 (Imp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ( ph  /\  ps )  ->  ch ) )
 
Theoremimpexp 259 Import-export theorem. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
 |-  ( ( ( ph  /\ 
 ps )  ->  ch )  <->  (
 ph  ->  ( ps  ->  ch ) ) )
 
Theorempm3.21 260 Join antecedents with conjunction. Theorem *3.21 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  ( ps  /\  ph ) ) )
 
Theorempm3.22 261 Theorem *3.22 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)
 |-  ( ( ph  /\  ps )  ->  ( ps  /\  ph ) )
 
Theoremancom 262 Commutative law for conjunction. Theorem *4.3 of [WhiteheadRussell] p. 118. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Wolf Lammen, 4-Nov-2012.)
 |-  ( ( ph  /\  ps ) 
 <->  ( ps  /\  ph )
 )
 
Theoremancomd 263 Commutation of conjuncts in consequent. (Contributed by Jeff Hankins, 14-Aug-2009.)
 |-  ( ph  ->  ( ps  /\  ch ) )   =>    |-  ( ph  ->  ( ch  /\ 
 ps ) )
 
Theoremancoms 264 Inference commuting conjunction in antecedent. (Contributed by NM, 21-Apr-1994.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ps  /\  ph )  ->  ch )
 
Theoremancomsd 265 Deduction commuting conjunction in antecedent. (Contributed by NM, 12-Dec-2004.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ph  ->  ( ( ch  /\  ps )  ->  th ) )
 
Theorempm3.2i 266 Infer conjunction of premises. (Contributed by NM, 5-Aug-1993.)
 |-  ph   &    |- 
 ps   =>    |-  ( ph  /\  ps )
 
Theorempm3.43i 267 Nested conjunction of antecedents. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( ph  ->  ps )  ->  ( ( ph  ->  ch )  ->  ( ph  ->  ( ps  /\  ch ) ) ) )
 
Theoremsimplbi 268 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
 |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ph  ->  ps )
 
Theoremsimprbi 269 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
 |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ph  ->  ch )
 
Theoremadantr 270 Inference adding a conjunct to the right of an antecedent. (Contributed by NM, 30-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ph  /\  ch )  ->  ps )
 
Theoremadantl 271 Inference adding a conjunct to the left of an antecedent. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ch  /\  ph )  ->  ps )
 
Theoremadantld 272 Deduction adding a conjunct to the left of an antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2012.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ( th  /\  ps )  ->  ch ) )
 
Theoremadantrd 273 Deduction adding a conjunct to the right of an antecedent. (Contributed by NM, 4-May-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ( ps  /\  th )  ->  ch ) )
 
Theoremimpel 274 An inference for implication elimination. (Contributed by Giovanni Mascellani, 23-May-2019.) (Proof shortened by Wolf Lammen, 2-Sep-2020.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ps )   =>    |-  ( ( ph  /\  th )  ->  ch )
 
Theoremmpan9 275 Modus ponens conjoining dissimilar antecedents. (Contributed by NM, 1-Feb-2008.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  ( ps  ->  th ) )   =>    |-  ( ( ph  /\  ch )  ->  th )
 
Theoremsyldan 276 A syllogism deduction with conjoined antecents. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 6-Apr-2013.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ps )  ->  th )
 
Theoremsylan 277 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
 |-  ( ph  ->  ps )   &    |-  (
 ( ps  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ch )  ->  th )
 
Theoremsylanb 278 A syllogism inference. (Contributed by NM, 18-May-1994.)
 |-  ( ph  <->  ps )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  (
 ( ph  /\  ch )  ->  th )
 
Theoremsylanbr 279 A syllogism inference. (Contributed by NM, 18-May-1994.)
 |-  ( ps  <->  ph )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  (
 ( ph  /\  ch )  ->  th )
 
Theoremsylan2 280 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
 |-  ( ph  ->  ch )   &    |-  (
 ( ps  /\  ch )  ->  th )   =>    |-  ( ( ps  /\  ph )  ->  th )
 
Theoremsylan2b 281 A syllogism inference. (Contributed by NM, 21-Apr-1994.)
 |-  ( ph  <->  ch )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  (
 ( ps  /\  ph )  ->  th )
 
Theoremsylan2br 282 A syllogism inference. (Contributed by NM, 21-Apr-1994.)
 |-  ( ch  <->  ph )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  (
 ( ps  /\  ph )  ->  th )
 
Theoremsyl2an 283 A double syllogism inference. (Contributed by NM, 31-Jan-1997.)
 |-  ( ph  ->  ps )   &    |-  ( ta  ->  ch )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  (
 ( ph  /\  ta )  ->  th )
 
Theoremsyl2anr 284 A double syllogism inference. (Contributed by NM, 17-Sep-2013.)
 |-  ( ph  ->  ps )   &    |-  ( ta  ->  ch )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  (
 ( ta  /\  ph )  ->  th )
 
Theoremsyl2anb 285 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
 |-  ( ph  <->  ps )   &    |-  ( ta  <->  ch )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  (
 ( ph  /\  ta )  ->  th )
 
Theoremsyl2anbr 286 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
 |-  ( ps  <->  ph )   &    |-  ( ch  <->  ta )   &    |-  ( ( ps 
 /\  ch )  ->  th )   =>    |-  (
 ( ph  /\  ta )  ->  th )
 
Theoremsyland 287 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ( ch 
 /\  th )  ->  ta )
 )   =>    |-  ( ph  ->  (
 ( ps  /\  th )  ->  ta ) )
 
Theoremsylan2d 288 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ( th  /\ 
 ch )  ->  ta )
 )   =>    |-  ( ph  ->  (
 ( th  /\  ps )  ->  ta ) )
 
Theoremsyl2and 289 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  ->  ta ) )   &    |-  ( ph  ->  ( ( ch  /\  ta )  ->  et ) )   =>    |-  ( ph  ->  ( ( ps  /\  th )  ->  et ) )
 
Theorembiimpa 290 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theorembiimpar 291 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ( ph  /\ 
 ch )  ->  ps )
 
Theorembiimpac 292 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ( ps 
 /\  ph )  ->  ch )
 
Theorembiimparc 293 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ( ch 
 /\  ph )  ->  ps )
 
Theoremiba 294 Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Mar-1994.) (Revised by NM, 24-Mar-2013.)
 |-  ( ph  ->  ( ps 
 <->  ( ps  /\  ph )
 ) )
 
Theoremibar 295 Introduction of antecedent as conjunct. (Contributed by NM, 5-Dec-1995.) (Revised by NM, 24-Mar-2013.)
 |-  ( ph  ->  ( ps 
 <->  ( ph  /\  ps ) ) )
 
Theorembiantru 296 A wff is equivalent to its conjunction with truth. (Contributed by NM, 5-Aug-1993.)
 |-  ph   =>    |-  ( ps  <->  ( ps  /\  ph ) )
 
Theorembiantrur 297 A wff is equivalent to its conjunction with truth. (Contributed by NM, 3-Aug-1994.)
 |-  ph   =>    |-  ( ps  <->  ( ph  /\  ps ) )
 
Theorembiantrud 298 A wff is equivalent to its conjunction with truth. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2013.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  <->  ( ch  /\  ps ) ) )
 
Theorembiantrurd 299 A wff is equivalent to its conjunction with truth. (Contributed by NM, 1-May-1995.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ch  <->  ( ps  /\  ch ) ) )
 
Theoremjca 300 Deduce conjunction of the consequents of two implications ("join consequents with 'and'"). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  ( ps  /\  ch ) )
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