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Theorem List for Metamath Proof Explorer - 401-500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempm2.68 401 Theorem *2.68 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)

Theoremdfor2 402 Logical 'or' expressed in terms of implication only. Theorem *5.25 of [WhiteheadRussell] p. 124. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Wolf Lammen, 20-Oct-2012.)

Theoremimor 403 Implication in terms of disjunction. Theorem *4.6 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.)

Theoremimori 404 Infer disjunction from implication. (Contributed by NM, 12-Mar-2012.)

Theoremimorri 405 Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremexmid 406 Law of excluded middle, also called the principle of tertium non datur. Theorem *2.11 of [WhiteheadRussell] p. 101. It says that something is either true or not true; there are no in-between values of truth. This is an essential distinction of our classical logic and is not a theorem of intuitionistic logic. (Contributed by NM, 5-Aug-1993.)

Theoremexmidd 407 Law of excluded middle in a context. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theorempm2.1 408 Theorem *2.1 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)

Theorempm2.13 409 Theorem *2.13 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)

Theorempm4.62 410 Theorem *4.62 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)

Theorempm4.66 411 Theorem *4.66 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)

Theorempm4.63 412 Theorem *4.63 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)

Theoremimnan 413 Express implication in terms of conjunction. (Contributed by NM, 9-Apr-1994.)

Theoremimnani 414 Express implication in terms of conjunction. (Contributed by Mario Carneiro, 28-Sep-2015.)

Theoremiman 415 Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2012.)

Theoremannim 416 Express conjunction in terms of implication. (Contributed by NM, 2-Aug-1994.)

Theorempm4.61 417 Theorem *4.61 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)

Theorempm4.65 418 Theorem *4.65 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)

Theorempm4.67 419 Theorem *4.67 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)

Theoremimp 420 Importation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)

Theoremimpcom 421 Importation inference with commuted antecedents. (Contributed by NM, 25-May-2005.)

Theoremimp3a 422 Importation deduction. (Contributed by NM, 31-Mar-1994.)

Theoremimp31 423 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremimp32 424 An importation inference. (Contributed by NM, 26-Apr-1994.)

Theoremex 425 Exportation inference. (This theorem used to be labeled "exp" but was changed to "ex" so as not to conflict with the math token "exp", per the June 2006 Metamath spec change.) A translation of natural deduction rule I ( introduction), see natded 4. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)

Theoremexpcom 426 Exportation inference with commuted antecedents. (Contributed by NM, 25-May-2005.)

Theoremexp3a 427 Exportation deduction. (Contributed by NM, 20-Aug-1993.)

Theoremexpdimp 428 A deduction version of exportation, followed by importation. (Contributed by NM, 6-Sep-2008.)

Theoremimpancom 429 Mixed importation/commutation inference. (Contributed by NM, 22-Jun-2013.)

Theoremcon3and 430 Variant of con3d 127 with importation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theorempm2.01da 431 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theorempm2.18da 432 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theorempm3.3 433 Theorem *3.3 (Exp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)

Theorempm3.31 434 Theorem *3.31 (Imp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)

Theoremimpexp 435 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.)

Theorempm3.2 436 Join antecedents with conjunction. Theorem *3.2 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)

Theorempm3.21 437 Join antecedents with conjunction. Theorem *3.21 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.)

Theorempm3.22 438 Theorem *3.22 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)

Theoremancom 439 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.)

Theoremancomd 440 Commutation of conjuncts in consequent. (Contributed by Jeff Hankins, 14-Aug-2009.)

Theoremancoms 441 Inference commuting conjunction in antecedent. (Contributed by NM, 21-Apr-1994.)

Theoremancomsd 442 Deduction commuting conjunction in antecedent. (Contributed by NM, 12-Dec-2004.)

Theorempm3.2i 443 Infer conjunction of premises. (Contributed by NM, 5-Aug-1993.)

Theorempm3.43i 444 Nested conjunction of antecedents. (Contributed by NM, 5-Aug-1993.)

Theoremsimpl 445 Elimination of a conjunct. Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)

Theoremsimpli 446 Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)

Theoremsimpld 447 Deduction eliminating a conjunct. A translation of natural deduction rule EL ( elimination left), see natded 4. (Contributed by NM, 5-Aug-1993.)

Theoremsimplbi 448 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)

Theoremsimpr 449 Elimination of a conjunct. Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)

Theoremsimpri 450 Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)

Theoremsimprd 451 Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.) A translation of natural deduction rule ER ( elimination right), see natded 4. (Proof shortened by Wolf Lammen, 3-Oct-2013.)

Theoremsimprbi 452 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)

Theoremadantr 453 Inference adding a conjunct to the right of an antecedent. (Contributed by NM, 30-Aug-1993.)

Theoremadantl 454 Inference adding a conjunct to the left of an antecedent. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)

Theoremadantld 455 Deduction adding a conjunct to the left of an antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2012.)

Theoremadantrd 456 Deduction adding a conjunct to the right of an antecedent. (Contributed by NM, 4-May-1994.)

Theoremmpan9 457 Modus ponens conjoining dissimilar antecedents. (Contributed by NM, 1-Feb-2008.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theoremsyldan 458 A syllogism deduction with conjoined antecedents. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 6-Apr-2013.)

Theoremsylan 459 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)

Theoremsylanb 460 A syllogism inference. (Contributed by NM, 18-May-1994.)

Theoremsylanbr 461 A syllogism inference. (Contributed by NM, 18-May-1994.)

Theoremsylan2 462 A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)

Theoremsylan2b 463 A syllogism inference. (Contributed by NM, 21-Apr-1994.)

Theoremsylan2br 464 A syllogism inference. (Contributed by NM, 21-Apr-1994.)

Theoremsyl2an 465 A double syllogism inference. (Contributed by NM, 31-Jan-1997.)

Theoremsyl2anr 466 A double syllogism inference. (Contributed by NM, 17-Sep-2013.)

Theoremsyl2anb 467 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)

Theoremsyl2anbr 468 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)

Theoremsyland 469 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)

Theoremsylan2d 470 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)

Theoremsyl2and 471 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)

Theorembiimpa 472 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)

Theorembiimpar 473 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)

Theorembiimpac 474 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)

Theorembiimparc 475 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)

Theoremianor 476 Negated conjunction in terms of disjunction (DeMorgan's law). Theorem *4.51 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremanor 477 Conjunction in terms of disjunction (DeMorgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Nov-2012.)

Theoremioran 478 Negated disjunction in terms of conjunction (DeMorgan's law). Compare Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theorempm4.52 479 Theorem *4.52 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Nov-2012.)

Theorempm4.53 480 Theorem *4.53 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)

Theorempm4.54 481 Theorem *4.54 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Nov-2012.)

Theorempm4.55 482 Theorem *4.55 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)

Theorempm4.56 483 Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)

Theoremoran 484 Disjunction in terms of conjunction (DeMorgan's law). Compare Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theorempm4.57 485 Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)

Theorempm3.1 486 Theorem *3.1 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)

Theorempm3.11 487 Theorem *3.11 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)

Theorempm3.12 488 Theorem *3.12 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)

Theorempm3.13 489 Theorem *3.13 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)

Theorempm3.14 490 Theorem *3.14 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)

Theoremiba 491 Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Mar-1994.)

Theoremibar 492 Introduction of antecedent as conjunct. (Contributed by NM, 5-Dec-1995.)

Theorembiantru 493 A wff is equivalent to its conjunction with truth. (Contributed by NM, 5-Aug-1993.)

Theorembiantrur 494 A wff is equivalent to its conjunction with truth. (Contributed by NM, 3-Aug-1994.)

Theorembiantrud 495 A wff is equivalent to its conjunction with truth. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2013.)

Theorembiantrurd 496 A wff is equivalent to its conjunction with truth. (Contributed by NM, 1-May-1995.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theoremjaao 497 Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)

Theoremjaoa 498 Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)

Theorempm3.44 499 Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)

Theoremjao 500 Disjunction of antecedents. Compare Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 4-Apr-2013.)

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