 Home Intuitionistic Logic ExplorerTheorem List (p. 4 of 95) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

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)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9457
 Copyright terms: Public domain < Previous  Next >