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

Theoremmpbir 201 An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)

Theoremmpbid 202 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)

Theoremmpbii 203 An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)

Theoremsylibr 204 A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by NM, 5-Aug-1993.)

Theoremsylbir 205 A mixed syllogism inference from a biconditional and an implication. (Contributed by NM, 5-Aug-1993.)

Theoremsylibd 206 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)

Theoremsylbid 207 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)

Theoremmpbidi 208 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.)

Theoremsyl5bi 209 A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded antecedent with a definition. (Contributed by NM, 5-Aug-1993.)

Theoremsyl5bir 210 A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)

Theoremsyl5ib 211 A mixed syllogism inference. (Contributed by NM, 5-Aug-1993.)

Theoremsyl5ibcom 212 A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.)

Theoremsyl5ibr 213 A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.)

Theoremsyl5ibrcom 214 A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.)

Theorembiimprd 215 Deduce a converse implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)

Theorembiimpcd 216 Deduce a commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)

Theorembiimprcd 217 Deduce a converse commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)

Theoremsyl6ib 218 A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)

Theoremsyl6ibr 219 A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded consequent with a definition. (Contributed by NM, 5-Aug-1993.)

Theoremsyl6bi 220 A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.)

Theoremsyl6bir 221 A mixed syllogism inference. (Contributed by NM, 18-May-1994.)

Theoremsyl7bi 222 A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)

Theoremsyl8ib 223 A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.)

Theoremmpbird 224 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)

Theoremmpbiri 225 An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)

Theoremsylibrd 226 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)

Theoremsylbird 227 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)

Theorembiid 228 Principle of identity for logical equivalence. Theorem *4.2 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.)

Theorembiidd 229 Principle of identity with antecedent. (Contributed by NM, 25-Nov-1995.)

Theorempm5.1im 230 Two propositions are equivalent if they are both true. Closed form of 2th 231. Equivalent to a bi1 179-like version of the xor-connective. This theorem stays true, no matter how you permute its operands. This is evident from its sharper version . (Contributed by Wolf Lammen, 12-May-2013.)

Theorem2th 231 Two truths are equivalent. (Contributed by NM, 18-Aug-1993.)

Theorem2thd 232 Two truths are equivalent (deduction rule). (Contributed by NM, 3-Jun-2012.)

Theoremibi 233 Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 17-Oct-2003.)

Theoremibir 234 Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 22-Jul-2004.)

Theoremibd 235 Deduction that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 26-Jun-2004.)

Theorempm5.74 236 Distribution of implication over biconditional. Theorem *5.74 of [WhiteheadRussell] p. 126. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 11-Apr-2013.)

Theorempm5.74i 237 Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.)

Theorempm5.74ri 238 Distribution of implication over biconditional (reverse inference rule). (Contributed by NM, 1-Aug-1994.)

Theorempm5.74d 239 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 21-Mar-1996.)

Theorempm5.74rd 240 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 19-Mar-1997.)

Theorembitri 241 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.)

Theorembitr2i 242 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theorembitr3i 243 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theorembitr4i 244 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theorembitrd 245 Deduction form of bitri 241. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.)

Theorembitr2d 246 Deduction form of bitr2i 242. (Contributed by NM, 9-Jun-2004.)

Theorembitr3d 247 Deduction form of bitr3i 243. (Contributed by NM, 5-Aug-1993.)

Theorembitr4d 248 Deduction form of bitr4i 244. (Contributed by NM, 5-Aug-1993.)

Theoremsyl5bb 249 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)

Theoremsyl5rbb 250 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)

Theoremsyl5bbr 251 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)

Theoremsyl5rbbr 252 A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)

Theoremsyl6bb 253 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)

Theoremsyl6rbb 254 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)

Theoremsyl6bbr 255 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)

Theoremsyl6rbbr 256 A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)

Theorem3imtr3i 257 A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994.)

Theorem3imtr4i 258 A mixed syllogism inference, useful for applying a definition to both sides of an implication. (Contributed by NM, 5-Aug-1993.)

Theorem3imtr3d 259 More general version of 3imtr3i 257. Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.)

Theorem3imtr4d 260 More general version of 3imtr4i 258. Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995.)

Theorem3imtr3g 261 More general version of 3imtr3i 257. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)

Theorem3imtr4g 262 More general version of 3imtr4i 258. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)

Theorem3bitri 263 A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theorem3bitrri 264 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)

Theorem3bitr2i 265 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)

Theorem3bitr2ri 266 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)

Theorem3bitr3i 267 A chained inference from transitive law for logical equivalence. (Contributed by NM, 19-Aug-1993.)

Theorem3bitr3ri 268 A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theorem3bitr4i 269 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 270 A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.)

Theorem3bitrd 271 Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999.)

Theorem3bitrrd 272 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)

Theorem3bitr2d 273 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)

Theorem3bitr2rd 274 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)

Theorem3bitr3d 275 Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 24-Apr-1996.)

Theorem3bitr3rd 276 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)

Theorem3bitr4d 277 Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 18-Oct-1995.)

Theorem3bitr4rd 278 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)

Theorem3bitr3g 279 More general version of 3bitr3i 267. Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.)

Theorem3bitr4g 280 More general version of 3bitr4i 269. Useful for converting definitions in a formula. (Contributed by NM, 5-Aug-1993.)

Theorembi3ant 281 Construct a bi-conditional in antecedent position. (Contributed by Wolf Lammen, 14-May-2013.)

Theorembisym 282 Express symmetries of theorems in terms of biconditionals. (Contributed by Wolf Lammen, 14-May-2013.)

Theoremnotnot 283 Double negation. Theorem *4.13 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.)

Theoremcon34b 284 Contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116. (Contributed by NM, 5-Aug-1993.)

Theoremcon4bid 285 A contraposition deduction. (Contributed by NM, 21-May-1994.)

Theoremnotbid 286 Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994.)

Theoremnotbi 287 Contraposition. Theorem *4.11 of [WhiteheadRussell] p. 117. (Contributed by NM, 21-May-1994.) (Proof shortened by Wolf Lammen, 12-Jun-2013.)

Theoremnotbii 288 Negate both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.)

Theoremcon4bii 289 A contraposition inference. (Contributed by NM, 21-May-1994.)

Theoremmtbi 290 An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)

Theoremmtbir 291 An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 14-Oct-2012.)

Theoremmtbid 292 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)

Theoremmtbird 293 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)

Theoremmtbii 294 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)

Theoremmtbiri 295 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)

Theoremsylnib 296 A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)

Theoremsylnibr 297 A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.)

Theoremsylnbi 298 A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.)

Theoremsylnbir 299 A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)

Theoremxchnxbi 300 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)

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