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

Theorem3bitr4i 201 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 202 A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.)
(φψ)    &   (χφ)    &   (θψ)       (θχ)

Theorem3bitrd 203 Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999.)
(φ → (ψχ))    &   (φ → (χθ))    &   (φ → (θτ))       (φ → (ψτ))

Theorem3bitrrd 204 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(φ → (ψχ))    &   (φ → (χθ))    &   (φ → (θτ))       (φ → (τψ))

Theorem3bitr2d 205 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(φ → (ψχ))    &   (φ → (θχ))    &   (φ → (θτ))       (φ → (ψτ))

Theorem3bitr2rd 206 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(φ → (ψχ))    &   (φ → (θχ))    &   (φ → (θτ))       (φ → (τψ))

Theorem3bitr3d 207 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 208 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(φ → (ψχ))    &   (φ → (ψθ))    &   (φ → (χτ))       (φ → (τθ))

Theorem3bitr4d 209 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 210 Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
(φ → (ψχ))    &   (φ → (θψ))    &   (φ → (τχ))       (φ → (τθ))

Theorem3bitr3g 211 More general version of 3bitr3i 199. Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.)
(φ → (ψχ))    &   (ψθ)    &   (χτ)       (φ → (θτ))

Theorem3bitr4g 212 More general version of 3bitr4i 201. Useful for converting definitions in a formula. (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))    &   (θψ)    &   (τχ)       (φ → (θτ))

Theorembi3ant 213 Construct a biconditional in antecedent position. (Contributed by Wolf Lammen, 14-May-2013.)
(φ → (ψχ))       (((θτ) → φ) → (((τθ) → ψ) → ((θτ) → χ)))

Theorembisym 214 Express symmetries of theorems in terms of biconditionals. (Contributed by Wolf Lammen, 14-May-2013.)
(((φψ) → (χθ)) → (((ψφ) → (θχ)) → ((φψ) → (χθ))))

Theoremimbi2i 215 Introduce an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 6-Feb-2013.)
(φψ)       ((χφ) ↔ (χψ))

Theorembibi2i 216 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.)
(φψ)       ((χφ) ↔ (χψ))

Theorembibi1i 217 Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)
(φψ)       ((φχ) ↔ (ψχ))

Theorembibi12i 218 The equivalence of two equivalences. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (χθ)       ((φχ) ↔ (ψθ))

Theoremimbi2d 219 Deduction adding an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))       (φ → ((θψ) ↔ (θχ)))

Theoremimbi1d 220 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.)
(φ → (ψχ))       (φ → ((ψθ) ↔ (χθ)))

Theorembibi2d 221 Deduction adding a biconditional to the left in an equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.)
(φ → (ψχ))       (φ → ((θψ) ↔ (θχ)))

Theorembibi1d 222 Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))       (φ → ((ψθ) ↔ (χθ)))

Theoremimbi12d 223 Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))    &   (φ → (θτ))       (φ → ((ψθ) ↔ (χτ)))

Theorembibi12d 224 Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))    &   (φ → (θτ))       (φ → ((ψθ) ↔ (χτ)))

Theoremimbi1 225 Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
((φψ) → ((φχ) ↔ (ψχ)))

Theoremimbi2 226 Theorem *4.85 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)
((φψ) → ((χφ) ↔ (χψ)))

Theoremimbi1i 227 Introduce a consequent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 17-Sep-2013.)
(φψ)       ((φχ) ↔ (ψχ))

Theoremimbi12i 228 Join two logical equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (χθ)       ((φχ) ↔ (ψθ))

Theorembibi1 229 Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
((φψ) → ((φχ) ↔ (ψχ)))

Theorembiimt 230 A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.)
(φ → (ψ ↔ (φψ)))

Theorempm5.5 231 Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
(φ → ((φψ) ↔ ψ))

Theorema1bi 232 Inference rule introducing a theorem as an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
φ       (ψ ↔ (φψ))

Theorempm5.501 233 Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 24-Jan-2013.)
(φ → (ψ ↔ (φψ)))

Theoremibib 234 Implication in terms of implication and biconditional. (Contributed by NM, 31-Mar-1994.) (Proof shortened by Wolf Lammen, 24-Jan-2013.)
((φψ) ↔ (φ → (φψ)))

Theoremibibr 235 Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.)
((φψ) ↔ (φ → (ψφ)))

Theoremtbt 236 A wff is equivalent to its equivalence with truth. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
φ       (ψ ↔ (ψφ))

Theorembi2.04 237 Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 5-Aug-1993.)
((φ → (ψχ)) ↔ (ψ → (φχ)))

Theorempm5.4 238 Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
((φ → (φψ)) ↔ (φψ))

Theoremimdi 239 Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
((φ → (ψχ)) ↔ ((φψ) → (φχ)))

Theorempm5.41 240 Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 12-Oct-2012.)
(((φψ) → (φχ)) ↔ (φ → (ψχ)))

Theoremimim21b 241 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.)
((ψφ) → (((φχ) → (ψθ)) ↔ (ψ → (χθ))))

Theoremimpd 242 Importation deduction. (Contributed by NM, 31-Mar-1994.)
(φ → (ψ → (χθ)))       (φ → ((ψ χ) → θ))

Theoremimp31 243 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χθ)))       (((φ ψ) χ) → θ)

Theoremimp32 244 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χθ)))       ((φ (ψ χ)) → θ)

Theoremexpd 245 Exportation deduction. (Contributed by NM, 20-Aug-1993.)
(φ → ((ψ χ) → θ))       (φ → (ψ → (χθ)))

Theoremexpdimp 246 A deduction version of exportation, followed by importation. (Contributed by NM, 6-Sep-2008.)
(φ → ((ψ χ) → θ))       ((φ ψ) → (χθ))

Theoremimpancom 247 Mixed importation/commutation inference. (Contributed by NM, 22-Jun-2013.)
((φ ψ) → (χθ))       ((φ χ) → (ψθ))

Theorempm3.3 248 Theorem *3.3 (Exp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
(((φ ψ) → χ) → (φ → (ψχ)))

Theorempm3.31 249 Theorem *3.31 (Imp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
((φ → (ψχ)) → ((φ ψ) → χ))

Theoremimpexp 250 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.)
(((φ ψ) → χ) ↔ (φ → (ψχ)))

Theorempm3.21 251 Join antecedents with conjunction. Theorem *3.21 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.)
(φ → (ψ → (ψ φ)))

Theorempm3.22 252 Theorem *3.22 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)
((φ ψ) → (ψ φ))

Theoremancom 253 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.)
((φ ψ) ↔ (ψ φ))

Theoremancomd 254 Commutation of conjuncts in consequent. (Contributed by Jeff Hankins, 14-Aug-2009.)
(φ → (ψ χ))       (φ → (χ ψ))

Theoremancoms 255 Inference commuting conjunction in antecedent. (Contributed by NM, 21-Apr-1994.)
((φ ψ) → χ)       ((ψ φ) → χ)

Theoremancomsd 256 Deduction commuting conjunction in antecedent. (Contributed by NM, 12-Dec-2004.)
(φ → ((ψ χ) → θ))       (φ → ((χ ψ) → θ))

Theorempm3.2i 257 Infer conjunction of premises. (Contributed by NM, 5-Aug-1993.)
φ    &   ψ       (φ ψ)

Theorempm3.43i 258 Nested conjunction of antecedents. (Contributed by NM, 5-Aug-1993.)
((φψ) → ((φχ) → (φ → (ψ χ))))

Theoremsimplbi 259 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
(φ ↔ (ψ χ))       (φψ)

Theoremsimprbi 260 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
(φ ↔ (ψ χ))       (φχ)

Theoremadantr 261 Inference adding a conjunct to the right of an antecedent. (Contributed by NM, 30-Aug-1993.)
(φψ)       ((φ χ) → ψ)

Theoremadantl 262 Inference adding a conjunct to the left of an antecedent. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
(φψ)       ((χ φ) → ψ)

Theoremadantld 263 Deduction adding a conjunct to the left of an antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2012.)
(φ → (ψχ))       (φ → ((θ ψ) → χ))

Theoremadantrd 264 Deduction adding a conjunct to the right of an antecedent. (Contributed by NM, 4-May-1994.)
(φ → (ψχ))       (φ → ((ψ θ) → χ))

Theoremmpan9 265 Modus ponens conjoining dissimilar antecedents. (Contributed by NM, 1-Feb-2008.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(φψ)    &   (χ → (ψθ))       ((φ χ) → θ)

Theoremsyldan 266 A syllogism deduction with conjoined antecents. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 6-Apr-2013.)
((φ ψ) → χ)    &   ((φ χ) → θ)       ((φ ψ) → θ)

Theoremsylan 267 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
(φψ)    &   ((ψ χ) → θ)       ((φ χ) → θ)

Theoremsylanb 268 A syllogism inference. (Contributed by NM, 18-May-1994.)
(φψ)    &   ((ψ χ) → θ)       ((φ χ) → θ)

Theoremsylanbr 269 A syllogism inference. (Contributed by NM, 18-May-1994.)
(ψφ)    &   ((ψ χ) → θ)       ((φ χ) → θ)

Theoremsylan2 270 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
(φχ)    &   ((ψ χ) → θ)       ((ψ φ) → θ)

Theoremsylan2b 271 A syllogism inference. (Contributed by NM, 21-Apr-1994.)
(φχ)    &   ((ψ χ) → θ)       ((ψ φ) → θ)

Theoremsylan2br 272 A syllogism inference. (Contributed by NM, 21-Apr-1994.)
(χφ)    &   ((ψ χ) → θ)       ((ψ φ) → θ)

Theoremsyl2an 273 A double syllogism inference. (Contributed by NM, 31-Jan-1997.)
(φψ)    &   (τχ)    &   ((ψ χ) → θ)       ((φ τ) → θ)

Theoremsyl2anr 274 A double syllogism inference. (Contributed by NM, 17-Sep-2013.)
(φψ)    &   (τχ)    &   ((ψ χ) → θ)       ((τ φ) → θ)

Theoremsyl2anb 275 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
(φψ)    &   (τχ)    &   ((ψ χ) → θ)       ((φ τ) → θ)

Theoremsyl2anbr 276 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
(ψφ)    &   (χτ)    &   ((ψ χ) → θ)       ((φ τ) → θ)

Theoremsyland 277 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
(φ → (ψχ))    &   (φ → ((χ θ) → τ))       (φ → ((ψ θ) → τ))

Theoremsylan2d 278 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
(φ → (ψχ))    &   (φ → ((θ χ) → τ))       (φ → ((θ ψ) → τ))

Theoremsyl2and 279 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
(φ → (ψχ))    &   (φ → (θτ))    &   (φ → ((χ τ) → η))       (φ → ((ψ θ) → η))

Theorembiimpa 280 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(φ → (ψχ))       ((φ ψ) → χ)

Theorembiimpar 281 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(φ → (ψχ))       ((φ χ) → ψ)

Theorembiimpac 282 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(φ → (ψχ))       ((ψ φ) → χ)

Theorembiimparc 283 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(φ → (ψχ))       ((χ φ) → ψ)

Theoremiba 284 Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Mar-1994.) (Revised by NM, 24-Mar-2013.)
(φ → (ψ ↔ (ψ φ)))

Theoremibar 285 Introduction of antecedent as conjunct. (Contributed by NM, 5-Dec-1995.) (Revised by NM, 24-Mar-2013.)
(φ → (ψ ↔ (φ ψ)))

Theorembiantru 286 A wff is equivalent to its conjunction with truth. (Contributed by NM, 5-Aug-1993.)
φ       (ψ ↔ (ψ φ))

Theorembiantrur 287 A wff is equivalent to its conjunction with truth. (Contributed by NM, 3-Aug-1994.)
φ       (ψ ↔ (φ ψ))

Theorembiantrud 288 A wff is equivalent to its conjunction with truth. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2013.)
(φψ)       (φ → (χ ↔ (χ ψ)))

Theorembiantrurd 289 A wff is equivalent to its conjunction with truth. (Contributed by NM, 1-May-1995.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(φψ)       (φ → (χ ↔ (ψ χ)))

Theoremjca 290 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.)
(φψ)    &   (φχ)       (φ → (ψ χ))

Theoremjcad 291 Deduction conjoining the consequents of two implications. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)
(φ → (ψχ))    &   (φ → (ψθ))       (φ → (ψ → (χ θ)))

Theoremjca31 292 Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.)
(φψ)    &   (φχ)    &   (φθ)       (φ → ((ψ χ) θ))

Theoremjca32 293 Join three consequents. (Contributed by FL, 1-Aug-2009.)
(φψ)    &   (φχ)    &   (φθ)       (φ → (ψ (χ θ)))

Theoremjcai 294 Deduction replacing implication with conjunction. (Contributed by NM, 5-Aug-1993.)
(φψ)    &   (φ → (ψχ))       (φ → (ψ χ))

Theoremjctil 295 Inference conjoining a theorem to left of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
(φψ)    &   χ       (φ → (χ ψ))

Theoremjctir 296 Inference conjoining a theorem to right of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
(φψ)    &   χ       (φ → (ψ χ))

Theoremjctl 297 Inference conjoining a theorem to the left of a consequent. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)
ψ       (φ → (ψ φ))

Theoremjctr 298 Inference conjoining a theorem to the right of a consequent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)
ψ       (φ → (φ ψ))

Theoremjctild 299 Deduction conjoining a theorem to left of consequent in an implication. (Contributed by NM, 21-Apr-2005.)
(φ → (ψχ))    &   (φθ)       (φ → (ψ → (θ χ)))

Theoremjctird 300 Deduction conjoining a theorem to right of consequent in an implication. (Contributed by NM, 21-Apr-2005.)
(φ → (ψχ))    &   (φθ)       (φ → (ψ → (χ θ)))

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