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Theorem List for Intuitionistic Logic Explorer - 201-300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem3bitr2i 201 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)

Theorem3bitr2ri 202 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)

Theorem3bitr3i 203 A chained inference from transitive law for logical equivalence. (Contributed by NM, 19-Aug-1993.)

Theorem3bitr3ri 204 A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theorem3bitr4i 205 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.)

Theorem3bitr4ri 206 A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.)

Theorem3bitrd 207 Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999.)

Theorem3bitrrd 208 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)

Theorem3bitr2d 209 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)

Theorem3bitr2rd 210 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)

Theorem3bitr3d 211 Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 24-Apr-1996.)

Theorem3bitr3rd 212 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)

Theorem3bitr4d 213 Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 18-Oct-1995.)

Theorem3bitr4rd 214 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)

Theorem3bitr3g 215 More general version of 3bitr3i 203. Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.)

Theorem3bitr4g 216 More general version of 3bitr4i 205. Useful for converting definitions in a formula. (Contributed by NM, 5-Aug-1993.)

Theorembi3ant 217 Construct a biconditional in antecedent position. (Contributed by Wolf Lammen, 14-May-2013.)

Theorembisym 218 Express symmetries of theorems in terms of biconditionals. (Contributed by Wolf Lammen, 14-May-2013.)

Theoremimbi2i 219 Introduce an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 6-Feb-2013.)

Theorembibi2i 220 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.)

Theorembibi1i 221 Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)

Theorembibi12i 222 The equivalence of two equivalences. (Contributed by NM, 5-Aug-1993.)

Theoremimbi2d 223 Deduction adding an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theoremimbi1d 224 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.)

Theorembibi2d 225 Deduction adding a biconditional to the left in an equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.)

Theorembibi1d 226 Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)

Theoremimbi12d 227 Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)

Theorembibi12d 228 Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 5-Aug-1993.)

Theoremimbi1 229 Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)

Theoremimbi2 230 Theorem *4.85 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)

Theoremimbi1i 231 Introduce a consequent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 17-Sep-2013.)

Theoremimbi12i 232 Join two logical equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)

Theorembibi1 233 Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)

Theorembiimt 234 A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.)

Theorempm5.5 235 Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)

Theorema1bi 236 Inference rule introducing a theorem as an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)

Theorempm5.501 237 Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 24-Jan-2013.)

Theoremibib 238 Implication in terms of implication and biconditional. (Contributed by NM, 31-Mar-1994.) (Proof shortened by Wolf Lammen, 24-Jan-2013.)

Theoremibibr 239 Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.)

Theoremtbt 240 A wff is equivalent to its equivalence with truth. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theorembi2.04 241 Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 5-Aug-1993.)

Theorempm5.4 242 Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)

Theoremimdi 243 Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)

Theorempm5.41 244 Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 12-Oct-2012.)

Theoremimim21b 245 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.)

Theoremimpd 246 Importation deduction. (Contributed by NM, 31-Mar-1994.)

Theoremimp31 247 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp32 248 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremexpd 249 Exportation deduction. (Contributed by NM, 20-Aug-1993.)

Theoremexpdimp 250 A deduction version of exportation, followed by importation. (Contributed by NM, 6-Sep-2008.)

Theoremimpancom 251 Mixed importation/commutation inference. (Contributed by NM, 22-Jun-2013.)

Theorempm3.3 252 Theorem *3.3 (Exp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)

Theorempm3.31 253 Theorem *3.31 (Imp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)

Theoremimpexp 254 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.)

Theorempm3.21 255 Join antecedents with conjunction. Theorem *3.21 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.)

Theorempm3.22 256 Theorem *3.22 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)

Theoremancom 257 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 258 Commutation of conjuncts in consequent. (Contributed by Jeff Hankins, 14-Aug-2009.)

Theoremancoms 259 Inference commuting conjunction in antecedent. (Contributed by NM, 21-Apr-1994.)

Theoremancomsd 260 Deduction commuting conjunction in antecedent. (Contributed by NM, 12-Dec-2004.)

Theorempm3.2i 261 Infer conjunction of premises. (Contributed by NM, 5-Aug-1993.)

Theorempm3.43i 262 Nested conjunction of antecedents. (Contributed by NM, 5-Aug-1993.)

Theoremsimplbi 263 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)

Theoremsimprbi 264 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)

Theoremadantr 265 Inference adding a conjunct to the right of an antecedent. (Contributed by NM, 30-Aug-1993.)

Theoremadantl 266 Inference adding a conjunct to the left of an antecedent. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)

Theoremadantld 267 Deduction adding a conjunct to the left of an antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2012.)

Theoremadantrd 268 Deduction adding a conjunct to the right of an antecedent. (Contributed by NM, 4-May-1994.)

Theoremmpan9 269 Modus ponens conjoining dissimilar antecedents. (Contributed by NM, 1-Feb-2008.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theoremsyldan 270 A syllogism deduction with conjoined antecents. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 6-Apr-2013.)

Theoremsylan 271 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)

Theoremsylanb 272 A syllogism inference. (Contributed by NM, 18-May-1994.)

Theoremsylanbr 273 A syllogism inference. (Contributed by NM, 18-May-1994.)

Theoremsylan2 274 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)

Theoremsylan2b 275 A syllogism inference. (Contributed by NM, 21-Apr-1994.)

Theoremsylan2br 276 A syllogism inference. (Contributed by NM, 21-Apr-1994.)

Theoremsyl2an 277 A double syllogism inference. (Contributed by NM, 31-Jan-1997.)

Theoremsyl2anr 278 A double syllogism inference. (Contributed by NM, 17-Sep-2013.)

Theoremsyl2anb 279 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)

Theoremsyl2anbr 280 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)

Theoremsyland 281 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)

Theoremsylan2d 282 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)

Theoremsyl2and 283 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)

Theorembiimpa 284 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)

Theorembiimpar 285 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)

Theorembiimpac 286 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)

Theorembiimparc 287 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)

Theoremiba 288 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 289 Introduction of antecedent as conjunct. (Contributed by NM, 5-Dec-1995.) (Revised by NM, 24-Mar-2013.)

Theorembiantru 290 A wff is equivalent to its conjunction with truth. (Contributed by NM, 5-Aug-1993.)

Theorembiantrur 291 A wff is equivalent to its conjunction with truth. (Contributed by NM, 3-Aug-1994.)

Theorembiantrud 292 A wff is equivalent to its conjunction with truth. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2013.)

Theorembiantrurd 293 A wff is equivalent to its conjunction with truth. (Contributed by NM, 1-May-1995.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theoremjca 294 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.)

Theoremjcad 295 Deduction conjoining the consequents of two implications. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)

Theoremjca31 296 Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.)

Theoremjca32 297 Join three consequents. (Contributed by FL, 1-Aug-2009.)

Theoremjcai 298 Deduction replacing implication with conjunction. (Contributed by NM, 5-Aug-1993.)

Theoremjctil 299 Inference conjoining a theorem to left of consequent in an implication. (Contributed by NM, 31-Dec-1993.)

Theoremjctir 300 Inference conjoining a theorem to right of consequent in an implication. (Contributed by NM, 31-Dec-1993.)

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