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Theorem List for Intuitionistic Logic Explorer - 301-400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremancl 301 Conjoin antecedent to left of consequent. (Contributed by NM, 15-Aug-1994.)
((φψ) → (φ → (φ ψ)))
 
Theoremanclb 302 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.)
((φψ) ↔ (φ → (φ ψ)))
 
Theorempm5.42 303 Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((φ → (ψχ)) ↔ (φ → (ψ → (φ χ))))
 
Theoremancr 304 Conjoin antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
((φψ) → (φ → (ψ φ)))
 
Theoremancrb 305 Conjoin antecedent to right of consequent. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
((φψ) ↔ (φ → (ψ φ)))
 
Theoremancli 306 Deduction conjoining antecedent to left of consequent. (Contributed by NM, 12-Aug-1993.)
(φψ)       (φ → (φ ψ))
 
Theoremancri 307 Deduction conjoining antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
(φψ)       (φ → (ψ φ))
 
Theoremancld 308 Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.)
(φ → (ψχ))       (φ → (ψ → (ψ χ)))
 
Theoremancrd 309 Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.)
(φ → (ψχ))       (φ → (ψ → (χ ψ)))
 
Theoremanc2l 310 Conjoin antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 14-Jul-2013.)
((φ → (ψχ)) → (φ → (ψ → (φ χ))))
 
Theoremanc2r 311 Conjoin antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.)
((φ → (ψχ)) → (φ → (ψ → (χ φ))))
 
Theoremanc2li 312 Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
(φ → (ψχ))       (φ → (ψ → (φ χ)))
 
Theoremanc2ri 313 Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
(φ → (ψχ))       (φ → (ψ → (χ φ)))
 
Theorempm3.41 314 Theorem *3.41 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((φχ) → ((φ ψ) → χ))
 
Theorempm3.42 315 Theorem *3.42 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((ψχ) → ((φ ψ) → χ))
 
Theorempm3.4 316 Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113. (Contributed by NM, 31-Jul-1995.)
((φ ψ) → (φψ))
 
Theorempm4.45im 317 Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.)
(φ ↔ (φ (ψφ)))
 
Theoremanim12d 318 Conjoin antecedents and consequents in a deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 18-Dec-2013.)
(φ → (ψχ))    &   (φ → (θτ))       (φ → ((ψ θ) → (χ τ)))
 
Theoremanim1d 319 Add a conjunct to right of antecedent and consequent in a deduction. (Contributed by NM, 3-Apr-1994.)
(φ → (ψχ))       (φ → ((ψ θ) → (χ θ)))
 
Theoremanim2d 320 Add a conjunct to left of antecedent and consequent in a deduction. (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))       (φ → ((θ ψ) → (θ χ)))
 
Theoremanim12i 321 Conjoin antecedents and consequents of two premises. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Dec-2013.)
(φψ)    &   (χθ)       ((φ χ) → (ψ θ))
 
Theoremanim12ci 322 Variant of anim12i 321 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(φψ)    &   (χθ)       ((φ χ) → (θ ψ))
 
Theoremanim1i 323 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
(φψ)       ((φ χ) → (ψ χ))
 
Theoremanim2i 324 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
(φψ)       ((χ φ) → (χ ψ))
 
Theoremanim12ii 325 Conjoin antecedents and consequents in a deduction. (Contributed by NM, 11-Nov-2007.) (Proof shortened by Wolf Lammen, 19-Jul-2013.)
(φ → (ψχ))    &   (θ → (ψτ))       ((φ θ) → (ψ → (χ τ)))
 
Theoremprth 326 Theorem *3.47 of [WhiteheadRussell] p. 113. It was proved by Leibniz, and it evidently pleased him enough to call it 'praeclarum theorema' (splendid theorem). (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
(((φψ) (χθ)) → ((φ χ) → (ψ θ)))
 
Theorempm3.33 327 Theorem *3.33 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
(((φψ) (ψχ)) → (φχ))
 
Theorempm3.34 328 Theorem *3.34 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
(((ψχ) (φψ)) → (φχ))
 
Theorempm3.35 329 Conjunctive detachment. Theorem *3.35 of [WhiteheadRussell] p. 112. (Contributed by NM, 14-Dec-2002.)
((φ (φψ)) → ψ)
 
Theorempm5.31 330 Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((χ (φψ)) → (φ → (ψ χ)))
 
Theoremimp4a 331 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χ → (θτ))))       (φ → (ψ → ((χ θ) → τ)))
 
Theoremimp4b 332 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χ → (θτ))))       ((φ ψ) → ((χ θ) → τ))
 
Theoremimp4c 333 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χ → (θτ))))       (φ → (((ψ χ) θ) → τ))
 
Theoremimp4d 334 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χ → (θτ))))       (φ → ((ψ (χ θ)) → τ))
 
Theoremimp41 335 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χ → (θτ))))       ((((φ ψ) χ) θ) → τ)
 
Theoremimp42 336 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χ → (θτ))))       (((φ (ψ χ)) θ) → τ)
 
Theoremimp43 337 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χ → (θτ))))       (((φ ψ) (χ θ)) → τ)
 
Theoremimp44 338 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χ → (θτ))))       ((φ ((ψ χ) θ)) → τ)
 
Theoremimp45 339 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χ → (θτ))))       ((φ (ψ (χ θ))) → τ)
 
Theoremimp5a 340 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(φ → (ψ → (χ → (θ → (τη)))))       (φ → (ψ → (χ → ((θ τ) → η))))
 
Theoremimp5d 341 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(φ → (ψ → (χ → (θ → (τη)))))       (((φ ψ) χ) → ((θ τ) → η))
 
Theoremimp5g 342 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(φ → (ψ → (χ → (θ → (τη)))))       ((φ ψ) → (((χ θ) τ) → η))
 
Theoremimp55 343 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(φ → (ψ → (χ → (θ → (τη)))))       (((φ (ψ (χ θ))) τ) → η)
 
Theoremimp511 344 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(φ → (ψ → (χ → (θ → (τη)))))       ((φ ((ψ (χ θ)) τ)) → η)
 
Theoremexpimpd 345 Exportation followed by a deduction version of importation. (Contributed by NM, 6-Sep-2008.)
((φ ψ) → (χθ))       (φ → ((ψ χ) → θ))
 
Theoremexp31 346 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((φ ψ) χ) → θ)       (φ → (ψ → (χθ)))
 
Theoremexp32 347 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((φ (ψ χ)) → θ)       (φ → (ψ → (χθ)))
 
Theoremexp4a 348 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → ((χ θ) → τ)))       (φ → (ψ → (χ → (θτ))))
 
Theoremexp4b 349 An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
((φ ψ) → ((χ θ) → τ))       (φ → (ψ → (χ → (θτ))))
 
Theoremexp4c 350 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (((ψ χ) θ) → τ))       (φ → (ψ → (χ → (θτ))))
 
Theoremexp4d 351 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(φ → ((ψ (χ θ)) → τ))       (φ → (ψ → (χ → (θτ))))
 
Theoremexp41 352 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((((φ ψ) χ) θ) → τ)       (φ → (ψ → (χ → (θτ))))
 
Theoremexp42 353 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((φ (ψ χ)) θ) → τ)       (φ → (ψ → (χ → (θτ))))
 
Theoremexp43 354 An exportation inference. (Contributed by NM, 26-Apr-1994.)
(((φ ψ) (χ θ)) → τ)       (φ → (ψ → (χ → (θτ))))
 
Theoremexp44 355 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((φ ((ψ χ) θ)) → τ)       (φ → (ψ → (χ → (θτ))))
 
Theoremexp45 356 An exportation inference. (Contributed by NM, 26-Apr-1994.)
((φ (ψ (χ θ))) → τ)       (φ → (ψ → (χ → (θτ))))
 
Theoremexpr 357 Export a wff from a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
((φ (ψ χ)) → θ)       ((φ ψ) → (χθ))
 
Theoremexp5c 358 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(φ → ((ψ χ) → ((θ τ) → η)))       (φ → (ψ → (χ → (θ → (τη)))))
 
Theoremexp53 359 An exportation inference. (Contributed by Jeff Hankins, 30-Aug-2009.)
((((φ ψ) (χ θ)) τ) → η)       (φ → (ψ → (χ → (θ → (τη)))))
 
Theoremexpl 360 Export a wff from a left conjunct. (Contributed by Jeff Hankins, 28-Aug-2009.)
(((φ ψ) χ) → θ)       (φ → ((ψ χ) → θ))
 
Theoremimpr 361 Import a wff into a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
((φ ψ) → (χθ))       ((φ (ψ χ)) → θ)
 
Theoremimpl 362 Export a wff from a left conjunct. (Contributed by Mario Carneiro, 9-Jul-2014.)
(φ → ((ψ χ) → θ))       (((φ ψ) χ) → θ)
 
Theoremimpac 363 Importation with conjunction in consequent. (Contributed by NM, 9-Aug-1994.)
(φ → (ψχ))       ((φ ψ) → (χ ψ))
 
Theoremexbiri 364 Inference form of exbir 1322. (Contributed by Alan Sare, 31-Dec-2011.) (Proof shortened by Wolf Lammen, 27-Jan-2013.)
((φ ψ) → (χθ))       (φ → (ψ → (θχ)))
 
Theoremsimprbda 365 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
(φ → (ψ ↔ (χ θ)))       ((φ ψ) → χ)
 
Theoremsimplbda 366 Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)
(φ → (ψ ↔ (χ θ)))       ((φ ψ) → θ)
 
Theoremsimplbi2 367 Deduction eliminating a conjunct. (Contributed by Alan Sare, 31-Dec-2011.)
(φ ↔ (ψ χ))       (ψ → (χφ))
 
Theoremdfbi2 368 A theorem similar to the standard definition of the biconditional. Definition of [Margaris] p. 49. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 31-Jan-2015.)
((φψ) ↔ ((φψ) (ψφ)))
 
Theorempm4.71 369 Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 2-Dec-2012.)
((φψ) ↔ (φ ↔ (φ ψ)))
 
Theorempm4.71r 370 Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM, 25-Jul-1999.)
((φψ) ↔ (φ ↔ (ψ φ)))
 
Theorempm4.71i 371 Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 4-Jan-2004.)
(φψ)       (φ ↔ (φ ψ))
 
Theorempm4.71ri 372 Inference converting an implication to a biconditional with conjunction. Inference from Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM, 1-Dec-2003.)
(φψ)       (φ ↔ (ψ φ))
 
Theorempm4.71d 373 Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by Mario Carneiro, 25-Dec-2016.)
(φ → (ψχ))       (φ → (ψ ↔ (ψ χ)))
 
Theorempm4.71rd 374 Deduction converting an implication to a biconditional with conjunction. Deduction from Theorem *4.71 of [WhiteheadRussell] p. 120. (Contributed by NM, 10-Feb-2005.)
(φ → (ψχ))       (φ → (ψ ↔ (χ ψ)))
 
Theorempm4.24 375 Theorem *4.24 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 14-Mar-2014.)
(φ ↔ (φ φ))
 
Theoremanidm 376 Idempotent law for conjunction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Mar-2014.)
((φ φ) ↔ φ)
 
Theoremanidms 377 Inference from idempotent law for conjunction. (Contributed by NM, 15-Jun-1994.)
((φ φ) → ψ)       (φψ)
 
Theoremanidmdbi 378 Conjunction idempotence with antecedent. (Contributed by Roy F. Longton, 8-Aug-2005.)
((φ → (ψ ψ)) ↔ (φψ))
 
Theoremanasss 379 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
(((φ ψ) χ) → θ)       ((φ (ψ χ)) → θ)
 
Theoremanassrs 380 Associative law for conjunction applied to antecedent (eliminates syllogism). (Contributed by NM, 15-Nov-2002.)
((φ (ψ χ)) → θ)       (((φ ψ) χ) → θ)
 
Theoremanass 381 Associative law for conjunction. Theorem *4.32 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
(((φ ψ) χ) ↔ (φ (ψ χ)))
 
Theoremsylanl1 382 A syllogism inference. (Contributed by NM, 10-Mar-2005.)
(φψ)    &   (((ψ χ) θ) → τ)       (((φ χ) θ) → τ)
 
Theoremsylanl2 383 A syllogism inference. (Contributed by NM, 1-Jan-2005.)
(φχ)    &   (((ψ χ) θ) → τ)       (((ψ φ) θ) → τ)
 
Theoremsylanr1 384 A syllogism inference. (Contributed by NM, 9-Apr-2005.)
(φχ)    &   ((ψ (χ θ)) → τ)       ((ψ (φ θ)) → τ)
 
Theoremsylanr2 385 A syllogism inference. (Contributed by NM, 9-Apr-2005.)
(φθ)    &   ((ψ (χ θ)) → τ)       ((ψ (χ φ)) → τ)
 
Theoremsylani 386 A syllogism inference. (Contributed by NM, 2-May-1996.)
(φχ)    &   (ψ → ((χ θ) → τ))       (ψ → ((φ θ) → τ))
 
Theoremsylan2i 387 A syllogism inference. (Contributed by NM, 1-Aug-1994.)
(φθ)    &   (ψ → ((χ θ) → τ))       (ψ → ((χ φ) → τ))
 
Theoremsyl2ani 388 A syllogism inference. (Contributed by NM, 3-Aug-1999.)
(φχ)    &   (ηθ)    &   (ψ → ((χ θ) → τ))       (ψ → ((φ η) → τ))
 
Theoremsylan9 389 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(φ → (ψχ))    &   (θ → (χτ))       ((φ θ) → (ψτ))
 
Theoremsylan9r 390 Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))    &   (θ → (χτ))       ((θ φ) → (ψτ))
 
Theoremsyl2anc 391 Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012.)
(φψ)    &   (φχ)    &   ((ψ χ) → θ)       (φθ)
 
Theoremsylancl 392 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
(φψ)    &   χ    &   ((ψ χ) → θ)       (φθ)
 
Theoremsylancr 393 Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.)
ψ    &   (φχ)    &   ((ψ χ) → θ)       (φθ)
 
Theoremsylanbrc 394 Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009.)
(φψ)    &   (φχ)    &   (θ ↔ (ψ χ))       (φθ)
 
Theoremsylancb 395 A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
(φψ)    &   (φχ)    &   ((ψ χ) → θ)       (φθ)
 
Theoremsylancbr 396 A syllogism inference combined with contraction. (Contributed by NM, 3-Sep-2004.)
(ψφ)    &   (χφ)    &   ((ψ χ) → θ)       (φθ)
 
Theoremsylancom 397 Syllogism inference with commutation of antecents. (Contributed by NM, 2-Jul-2008.)
((φ ψ) → χ)    &   ((χ ψ) → θ)       ((φ ψ) → θ)
 
Theoremmpdan 398 An inference based on modus ponens. (Contributed by NM, 23-May-1999.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
(φψ)    &   ((φ ψ) → χ)       (φχ)
 
Theoremmpancom 399 An inference based on modus ponens with commutation of antecedents. (Contributed by NM, 28-Oct-2003.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
(ψφ)    &   ((φ ψ) → χ)       (ψχ)
 
Theoremmpan 400 An inference based on modus ponens. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)
φ    &   ((φ ψ) → χ)       (ψχ)
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