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Theorem List for Intuitionistic Logic Explorer - 201-300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem3bitrd 201 Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999.)
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Theorem3bitrrd 202 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
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Theorem3bitr2d 203 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
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Theorem3bitr2rd 204 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
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Theorem3bitr3d 205 Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 24-Apr-1996.)
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Theorem3bitr3rd 206 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
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Theorem3bitr4d 207 Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 18-Oct-1995.)
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Theorem3bitr4rd 208 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
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Theorem3bitr3g 209 More general version of 3bitr3i 197. Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.)
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Theorem3bitr4g 210 More general version of 3bitr4i 199. Useful for converting definitions in a formula. (Contributed by NM, 5-Aug-1993.)
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Theorembi3ant 211 Construct a bi-conditional in antecedent position. (Contributed by Wolf Lammen, 14-May-2013.)
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Theorembisym 212 Express symmetries of theorems in terms of biconditionals. (Contributed by Wolf Lammen, 14-May-2013.)
 
Theoremimbi2i 213 Introduce an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 6-Feb-2013.)
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Theorembibi2i 214 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.)
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Theorembibi1i 215 Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)
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Theorembibi12i 216 The equivalence of two equivalences. (Contributed by NM, 5-Aug-1993.)
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Theoremimbi2d 217 Deduction adding an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
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Theoremimbi1d 218 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.)
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Theorembibi2d 219 Deduction adding a biconditional to the left in an equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.)
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Theorembibi1d 220 Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)
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Theoremimbi12d 221 Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)
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Theorembibi12d 222 Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 5-Aug-1993.)
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Theoremimbi1 223 Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
 
Theoremimbi2 224 Theorem *4.85 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)
 
Theoremimbi1i 225 Introduce a consequent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 17-Sep-2013.)
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Theoremimbi12i 226 Join two logical equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)
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Theorembibi1 227 Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
 
Theorembiimt 228 A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.)
 
Theorempm5.5 229 Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 
Theorema1bi 230 Inference rule introducing a theorem as an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
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Theorempm5.501 231 Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 24-Jan-2013.)
 
Theoremibib 232 Implication in terms of implication and biconditional. (Contributed by NM, 31-Mar-1994.) (Proof shortened by Wolf Lammen, 24-Jan-2013.)
 
Theoremibibr 233 Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.)
 
Theoremtbt 234 A wff is equivalent to its equivalence with truth. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
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Theorembi2.04 235 Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 5-Aug-1993.)
 
Theorempm5.4 236 Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
 
Theoremimdi 237 Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
 
Theorempm5.41 238 Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 12-Oct-2012.)
 
Theoremimim21b 239 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.)
 
Theoremimp3a 240 Importation deduction. (Contributed by NM, 31-Mar-1994.) (Revised by NM, 24-Mar-2013.)
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Theoremimp31 241 An importation inference. (Contributed by NM, 26-Apr-1994.)
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Theoremimp32 242 An importation inference. (Contributed by NM, 26-Apr-1994.)
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Theoremexp3a 243 Exportation deduction. (Contributed by NM, 20-Aug-1993.) (Revised by NM, 24-Mar-2013.)
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Theoremexpdimp 244 A deduction version of exportation, followed by importation. (Contributed by NM, 6-Sep-2008.)
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Theoremimpancom 245 Mixed importation/commutation inference. (Contributed by NM, 22-Jun-2013.)
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Theorempm3.3 246 Theorem *3.3 (Exp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
 
Theorempm3.31 247 Theorem *3.31 (Imp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
 
Theoremimpexp 248 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.)
 
Theorempm3.21 249 Join antecedents with conjunction. Theorem *3.21 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.)
 
Theorempm3.22 250 Theorem *3.22 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)
 
Theoremancom 251 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.)
 
Theoremancomd 252 Commutation of conjuncts in consequent. (Contributed by Jeff Hankins, 14-Aug-2009.)
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Theoremancoms 253 Inference commuting conjunction in antecedent. (Contributed by NM, 21-Apr-1994.)
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Theoremancomsd 254 Deduction commuting conjunction in antecedent. (Contributed by NM, 12-Dec-2004.)
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Theorempm3.2i 255 Infer conjunction of premises. (Contributed by NM, 5-Aug-1993.)
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Theorempm3.43i 256 Nested conjunction of antecedents. (Contributed by NM, 5-Aug-1993.)
 
Theoremsimplbi 257 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
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Theoremsimprbi 258 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
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Theoremadantr 259 Inference adding a conjunct to the right of an antecedent. (Contributed by NM, 30-Aug-1993.)
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Theoremadantl 260 Inference adding a conjunct to the left of an antecedent. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
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Theoremadantld 261 Deduction adding a conjunct to the left of an antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2012.)
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Theoremadantrd 262 Deduction adding a conjunct to the right of an antecedent. (Contributed by NM, 4-May-1994.)
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Theoremmpan9 263 Modus ponens conjoining dissimilar antecedents. (Contributed by NM, 1-Feb-2008.) (Proof shortened by Andrew Salmon, 7-May-2011.)
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Theoremsyldan 264 A syllogism deduction with conjoined antecents. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 6-Apr-2013.)
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Theoremsylan 265 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
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Theoremsylanb 266 A syllogism inference. (Contributed by NM, 18-May-1994.)
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Theoremsylanbr 267 A syllogism inference. (Contributed by NM, 18-May-1994.)
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Theoremsylan2 268 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
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Theoremsylan2b 269 A syllogism inference. (Contributed by NM, 21-Apr-1994.)
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Theoremsylan2br 270 A syllogism inference. (Contributed by NM, 21-Apr-1994.)
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Theoremsyl2an 271 A double syllogism inference. (Contributed by NM, 31-Jan-1997.)
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Theoremsyl2anr 272 A double syllogism inference. (Contributed by NM, 17-Sep-2013.)
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Theoremsyl2anb 273 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
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Theoremsyl2anbr 274 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
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Theoremsyland 275 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
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Theoremsylan2d 276 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
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Theoremsyl2and 277 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
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Theorembiimpa 278 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
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Theorembiimpar 279 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
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Theorembiimpac 280 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
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Theorembiimparc 281 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
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Theoremiba 282 Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Mar-1994.) (Revised by NM, 24-Mar-2013.)
 
Theoremibar 283 Introduction of antecedent as conjunct. (Contributed by NM, 5-Dec-1995.) (Revised by NM, 24-Mar-2013.)
 
Theorembiantru 284 A wff is equivalent to its conjunction with truth. (Contributed by NM, 5-Aug-1993.)
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Theorembiantrur 285 A wff is equivalent to its conjunction with truth. (Contributed by NM, 3-Aug-1994.)
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Theorembiantrud 286 A wff is equivalent to its conjunction with truth. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2013.)
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Theorembiantrurd 287 A wff is equivalent to its conjunction with truth. (Contributed by NM, 1-May-1995.) (Proof shortened by Andrew Salmon, 7-May-2011.)
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Theoremjca 288 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.)
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Theoremjcad 289 Deduction conjoining the consequents of two implications. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)
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Theoremjca31 290 Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.)
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Theoremjca32 291 Join three consequents. (Contributed by FL, 1-Aug-2009.)
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Theoremjcai 292 Deduction replacing implication with conjunction. (Contributed by NM, 5-Aug-1993.)
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Theoremjctil 293 Inference conjoining a theorem to left of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
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Theoremjctir 294 Inference conjoining a theorem to right of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
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Theoremjctl 295 Inference conjoining a theorem to the left of a consequent. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)
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Theoremjctr 296 Inference conjoining a theorem to the right of a consequent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)
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Theoremjctild 297 Deduction conjoining a theorem to left of consequent in an implication. (Contributed by NM, 21-Apr-2005.)
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Theoremjctird 298 Deduction conjoining a theorem to right of consequent in an implication. (Contributed by NM, 21-Apr-2005.)
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Theoremancl 299 Conjoin antecedent to left of consequent. (Contributed by NM, 15-Aug-1994.)
 
Theoremanclb 300 Conjoin antecedent to left of consequent. Theorem *4.7 of [WhiteheadRussell] p. 120. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
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