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Theorem List for New Foundations Explorer - 201-300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmpbid 201 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (φ → (ψχ))       (φχ)
 
Theoremmpbii 202 An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
ψ    &   (φ → (ψχ))       (φχ)
 
Theoremsylibr 203 A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (χψ)       (φχ)
 
Theoremsylbir 204 A mixed syllogism inference from a biconditional and an implication. (Contributed by NM, 5-Aug-1993.)
(ψφ)    &   (ψχ)       (φχ)
 
Theoremsylibd 205 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
(φ → (ψχ))    &   (φ → (χθ))       (φ → (ψθ))
 
Theoremsylbid 206 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
(φ → (ψχ))    &   (φ → (χθ))       (φ → (ψθ))
 
Theoremmpbidi 207 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.)
(θ → (φψ))    &   (φ → (ψχ))       (θ → (φχ))
 
Theoremsyl5bi 208 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.)
(φψ)    &   (χ → (ψθ))       (χ → (φθ))
 
Theoremsyl5bir 209 A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)
(ψφ)    &   (χ → (ψθ))       (χ → (φθ))
 
Theoremsyl5ib 210 A mixed syllogism inference. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (χ → (ψθ))       (χ → (φθ))
 
Theoremsyl5ibcom 211 A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.)
(φψ)    &   (χ → (ψθ))       (φ → (χθ))
 
Theoremsyl5ibr 212 A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.)
(φθ)    &   (χ → (ψθ))       (χ → (φψ))
 
Theoremsyl5ibrcom 213 A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.)
(φθ)    &   (χ → (ψθ))       (φ → (χψ))
 
Theorembiimprd 214 Deduce a converse implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
(φ → (ψχ))       (φ → (χψ))
 
Theorembiimpcd 215 Deduce a commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
(φ → (ψχ))       (ψ → (φχ))
 
Theorembiimprcd 216 Deduce a converse commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
(φ → (ψχ))       (χ → (φψ))
 
Theoremsyl6ib 217 A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))    &   (χθ)       (φ → (ψθ))
 
Theoremsyl6ibr 218 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.)
(φ → (ψχ))    &   (θχ)       (φ → (ψθ))
 
Theoremsyl6bi 219 A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.)
(φ → (ψχ))    &   (χθ)       (φ → (ψθ))
 
Theoremsyl6bir 220 A mixed syllogism inference. (Contributed by NM, 18-May-1994.)
(φ → (χψ))    &   (χθ)       (φ → (ψθ))
 
Theoremsyl7bi 221 A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (χ → (θ → (ψτ)))       (χ → (θ → (φτ)))
 
Theoremsyl8ib 222 A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.)
(φ → (ψ → (χθ)))    &   (θτ)       (φ → (ψ → (χτ)))
 
Theoremmpbird 223 A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
(φχ)    &   (φ → (ψχ))       (φψ)
 
Theoremmpbiri 224 An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
χ    &   (φ → (ψχ))       (φψ)
 
Theoremsylibrd 225 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
(φ → (ψχ))    &   (φ → (θχ))       (φ → (ψθ))
 
Theoremsylbird 226 A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
(φ → (χψ))    &   (φ → (χθ))       (φ → (ψθ))
 
Theorembiid 227 Principle of identity for logical equivalence. Theorem *4.2 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.)
(φφ)
 
Theorembiidd 228 Principle of identity with antecedent. (Contributed by NM, 25-Nov-1995.)
(φ → (ψψ))
 
Theorempm5.1im 229 Two propositions are equivalent if they are both true. Closed form of 2th 230. Equivalent to a bi1 178-like version of the xor-connective. This theorem stays true, no matter how you permute its operands. This is evident from its sharper version (φ ↔ (ψ ↔ (φψ))). (Contributed by Wolf Lammen, 12-May-2013.)
(φ → (ψ → (φψ)))
 
Theorem2th 230 Two truths are equivalent. (Contributed by NM, 18-Aug-1993.)
φ    &   ψ       (φψ)
 
Theorem2thd 231 Two truths are equivalent (deduction rule). (Contributed by NM, 3-Jun-2012.)
(φψ)    &   (φχ)       (φ → (ψχ))
 
Theoremibi 232 Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 17-Oct-2003.)
(φ → (φψ))       (φψ)
 
Theoremibir 233 Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 22-Jul-2004.)
(φ → (ψφ))       (φψ)
 
Theoremibd 234 Deduction that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 26-Jun-2004.)
(φ → (ψ → (ψχ)))       (φ → (ψχ))
 
Theorempm5.74 235 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.)
((φ → (ψχ)) ↔ ((φψ) ↔ (φχ)))
 
Theorempm5.74i 236 Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.)
(φ → (ψχ))       ((φψ) ↔ (φχ))
 
Theorempm5.74ri 237 Distribution of implication over biconditional (reverse inference rule). (Contributed by NM, 1-Aug-1994.)
((φψ) ↔ (φχ))       (φ → (ψχ))
 
Theorempm5.74d 238 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 21-Mar-1996.)
(φ → (ψ → (χθ)))       (φ → ((ψχ) ↔ (ψθ)))
 
Theorempm5.74rd 239 Distribution of implication over biconditional (deduction rule). (Contributed by NM, 19-Mar-1997.)
(φ → ((ψχ) ↔ (ψθ)))       (φ → (ψ → (χθ)))
 
Theorembitri 240 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.)
(φψ)    &   (ψχ)       (φχ)
 
Theorembitr2i 241 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (ψχ)       (χφ)
 
Theorembitr3i 242 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
(ψφ)    &   (ψχ)       (φχ)
 
Theorembitr4i 243 An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (χψ)       (φχ)
 
Theorembitrd 244 Deduction form of bitri 240. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.)
(φ → (ψχ))    &   (φ → (χθ))       (φ → (ψθ))
 
Theorembitr2d 245 Deduction form of bitr2i 241. (Contributed by NM, 9-Jun-2004.)
(φ → (ψχ))    &   (φ → (χθ))       (φ → (θψ))
 
Theorembitr3d 246 Deduction form of bitr3i 242. (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))    &   (φ → (ψθ))       (φ → (χθ))
 
Theorembitr4d 247 Deduction form of bitr4i 243. (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))    &   (φ → (θχ))       (φ → (ψθ))
 
Theoremsyl5bb 248 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (χ → (ψθ))       (χ → (φθ))
 
Theoremsyl5rbb 249 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (χ → (ψθ))       (χ → (θφ))
 
Theoremsyl5bbr 250 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
(ψφ)    &   (χ → (ψθ))       (χ → (φθ))
 
Theoremsyl5rbbr 251 A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
(ψφ)    &   (χ → (ψθ))       (χ → (θφ))
 
Theoremsyl6bb 252 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))    &   (χθ)       (φ → (ψθ))
 
Theoremsyl6rbb 253 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))    &   (χθ)       (φ → (θψ))
 
Theoremsyl6bbr 254 A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))    &   (θχ)       (φ → (ψθ))
 
Theoremsyl6rbbr 255 A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
(φ → (ψχ))    &   (θχ)       (φ → (θψ))
 
Theorem3imtr3i 256 A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994.)
(φψ)    &   (φχ)    &   (ψθ)       (χθ)
 
Theorem3imtr4i 257 A mixed syllogism inference, useful for applying a definition to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (χφ)    &   (θψ)       (χθ)
 
Theorem3imtr3d 258 More general version of 3imtr3i 256. Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.)
(φ → (ψχ))    &   (φ → (ψθ))    &   (φ → (χτ))       (φ → (θτ))
 
Theorem3imtr4d 259 More general version of 3imtr4i 257. Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995.)
(φ → (ψχ))    &   (φ → (θψ))    &   (φ → (τχ))       (φ → (θτ))
 
Theorem3imtr3g 260 More general version of 3imtr3i 256. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
(φ → (ψχ))    &   (ψθ)    &   (χτ)       (φ → (θτ))
 
Theorem3imtr4g 261 More general version of 3imtr4i 257. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
(φ → (ψχ))    &   (θψ)    &   (τχ)       (φ → (θτ))
 
Theorem3bitri 262 A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (ψχ)    &   (χθ)       (φθ)
 
Theorem3bitrri 263 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
(φψ)    &   (ψχ)    &   (χθ)       (θφ)
 
Theorem3bitr2i 264 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
(φψ)    &   (χψ)    &   (χθ)       (φθ)
 
Theorem3bitr2ri 265 A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
(φψ)    &   (χψ)    &   (χθ)       (θφ)
 
Theorem3bitr3i 266 A chained inference from transitive law for logical equivalence. (Contributed by NM, 19-Aug-1993.)
(φψ)    &   (φχ)    &   (ψθ)       (χθ)
 
Theorem3bitr3ri 267 A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (φχ)    &   (ψθ)       (θχ)
 
Theorem3bitr4i 268 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.)
(φψ)    &   (χφ)    &   (θψ)       (χθ)
 
Theorem3bitr4ri 269 A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.)
(φψ)    &   (χφ)    &   (θψ)       (θχ)
 
Theorem3bitrd 270 Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999.)
(φ → (ψχ))    &   (φ → (χθ))    &   (φ → (θτ))       (φ → (ψτ))
 
Theorem3bitrrd 271 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(φ → (ψχ))    &   (φ → (χθ))    &   (φ → (θτ))       (φ → (τψ))
 
Theorem3bitr2d 272 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(φ → (ψχ))    &   (φ → (θχ))    &   (φ → (θτ))       (φ → (ψτ))
 
Theorem3bitr2rd 273 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(φ → (ψχ))    &   (φ → (θχ))    &   (φ → (θτ))       (φ → (τψ))
 
Theorem3bitr3d 274 Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 24-Apr-1996.)
(φ → (ψχ))    &   (φ → (ψθ))    &   (φ → (χτ))       (φ → (θτ))
 
Theorem3bitr3rd 275 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(φ → (ψχ))    &   (φ → (ψθ))    &   (φ → (χτ))       (φ → (τθ))
 
Theorem3bitr4d 276 Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 18-Oct-1995.)
(φ → (ψχ))    &   (φ → (θψ))    &   (φ → (τχ))       (φ → (θτ))
 
Theorem3bitr4rd 277 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(φ → (ψχ))    &   (φ → (θψ))    &   (φ → (τχ))       (φ → (τθ))
 
Theorem3bitr3g 278 More general version of 3bitr3i 266. Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.)
(φ → (ψχ))    &   (ψθ)    &   (χτ)       (φ → (θτ))
 
Theorem3bitr4g 279 More general version of 3bitr4i 268. Useful for converting definitions in a formula. (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))    &   (θψ)    &   (τχ)       (φ → (θτ))
 
Theorembi3ant 280 Construct a bi-conditional in antecedent position. (Contributed by Wolf Lammen, 14-May-2013.)
(φ → (ψχ))       (((θτ) → φ) → (((τθ) → ψ) → ((θτ) → χ)))
 
Theorembisym 281 Express symmetries of theorems in terms of biconditionals. (Contributed by Wolf Lammen, 14-May-2013.)
(((φψ) → (χθ)) → (((ψφ) → (θχ)) → ((φψ) → (χθ))))
 
Theoremnotnot 282 Double negation. Theorem *4.13 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.)
(φ ↔ ¬ ¬ φ)
 
Theoremcon34b 283 Contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116. (Contributed by NM, 5-Aug-1993.)
((φψ) ↔ (¬ ψ → ¬ φ))
 
Theoremcon4bid 284 A contraposition deduction. (Contributed by NM, 21-May-1994.)
(φ → (¬ ψ ↔ ¬ χ))       (φ → (ψχ))
 
Theoremnotbid 285 Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994.)
(φ → (ψχ))       (φ → (¬ ψ ↔ ¬ χ))
 
Theoremnotbi 286 Contraposition. Theorem *4.11 of [WhiteheadRussell] p. 117. (Contributed by NM, 21-May-1994.) (Proof shortened by Wolf Lammen, 12-Jun-2013.)
((φψ) ↔ (¬ φ ↔ ¬ ψ))
 
Theoremnotbii 287 Negate both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.)
(φψ)       φ ↔ ¬ ψ)
 
Theoremcon4bii 288 A contraposition inference. (Contributed by NM, 21-May-1994.)
φ ↔ ¬ ψ)       (φψ)
 
Theoremmtbi 289 An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
¬ φ    &   (φψ)        ¬ ψ
 
Theoremmtbir 290 An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 14-Oct-2012.)
¬ ψ    &   (φψ)        ¬ φ
 
Theoremmtbid 291 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.)
(φ → ¬ ψ)    &   (φ → (ψχ))       (φ → ¬ χ)
 
Theoremmtbird 292 A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.)
(φ → ¬ χ)    &   (φ → (ψχ))       (φ → ¬ ψ)
 
Theoremmtbii 293 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.)
¬ ψ    &   (φ → (ψχ))       (φ → ¬ χ)
 
Theoremmtbiri 294 An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.)
¬ χ    &   (φ → (ψχ))       (φ → ¬ ψ)
 
Theoremsylnib 295 A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)
(φ → ¬ ψ)    &   (ψχ)       (φ → ¬ χ)
 
Theoremsylnibr 296 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.)
(φ → ¬ ψ)    &   (χψ)       (φ → ¬ χ)
 
Theoremsylnbi 297 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.)
(φψ)    &   ψχ)       φχ)
 
Theoremsylnbir 298 A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.)
(ψφ)    &   ψχ)       φχ)
 
Theoremxchnxbi 299 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
φψ)    &   (φχ)       χψ)
 
Theoremxchnxbir 300 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)
φψ)    &   (χφ)       χψ)
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