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Theorem List for New Foundations Explorer - 401-500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremimor 401 Implication in terms of disjunction. Theorem *4.6 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.)
((φψ) ↔ (¬ φ ψ))
 
Theoremimori 402 Infer disjunction from implication. (Contributed by NM, 12-Mar-2012.)
(φψ)       φ ψ)
 
Theoremimorri 403 Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
φ ψ)       (φψ)
 
Theoremexmid 404 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.)
(φ ¬ φ)
 
Theoremexmidd 405 Law of excluded middle in a context. (Contributed by Mario Carneiro, 9-Feb-2017.)
(φ → (ψ ¬ ψ))
 
Theorempm2.1 406 Theorem *2.1 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
φ φ)
 
Theorempm2.13 407 Theorem *2.13 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
(φ ¬ ¬ ¬ φ)
 
Theorempm4.62 408 Theorem *4.62 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
((φ → ¬ ψ) ↔ (¬ φ ¬ ψ))
 
Theorempm4.66 409 Theorem *4.66 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
((¬ φ → ¬ ψ) ↔ (φ ¬ ψ))
 
Theorempm4.63 410 Theorem *4.63 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (φ → ¬ ψ) ↔ (φ ψ))
 
Theoremimnan 411 Express implication in terms of conjunction. (Contributed by NM, 9-Apr-1994.)
((φ → ¬ ψ) ↔ ¬ (φ ψ))
 
Theoremimnani 412 Express implication in terms of conjunction. (Contributed by Mario Carneiro, 28-Sep-2015.)
¬ (φ ψ)       (φ → ¬ ψ)
 
Theoremiman 413 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.)
((φψ) ↔ ¬ (φ ¬ ψ))
 
Theoremannim 414 Express conjunction in terms of implication. (Contributed by NM, 2-Aug-1994.)
((φ ¬ ψ) ↔ ¬ (φψ))
 
Theorempm4.61 415 Theorem *4.61 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (φψ) ↔ (φ ¬ ψ))
 
Theorempm4.65 416 Theorem *4.65 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (¬ φψ) ↔ (¬ φ ¬ ψ))
 
Theorempm4.67 417 Theorem *4.67 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (¬ φ → ¬ ψ) ↔ (¬ φ ψ))
 
Theoremimp 418 Importation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
(φ → (ψχ))       ((φ ψ) → χ)
 
Theoremimpcom 419 Importation inference with commuted antecedents. (Contributed by NM, 25-May-2005.)
(φ → (ψχ))       ((ψ φ) → χ)
 
Theoremimp3a 420 Importation deduction. (Contributed by NM, 31-Mar-1994.)
(φ → (ψ → (χθ)))       (φ → ((ψ χ) → θ))
 
Theoremimp31 421 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χθ)))       (((φ ψ) χ) → θ)
 
Theoremimp32 422 An importation inference. (Contributed by NM, 26-Apr-1994.)
(φ → (ψ → (χθ)))       ((φ (ψ χ)) → θ)
 
Theoremex 423 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 in set.mm. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
((φ ψ) → χ)       (φ → (ψχ))
 
Theoremexpcom 424 Exportation inference with commuted antecedents. (Contributed by NM, 25-May-2005.)
((φ ψ) → χ)       (ψ → (φχ))
 
Theoremexp3a 425 Exportation deduction. (Contributed by NM, 20-Aug-1993.)
(φ → ((ψ χ) → θ))       (φ → (ψ → (χθ)))
 
Theoremexpdimp 426 A deduction version of exportation, followed by importation. (Contributed by NM, 6-Sep-2008.)
(φ → ((ψ χ) → θ))       ((φ ψ) → (χθ))
 
Theoremimpancom 427 Mixed importation/commutation inference. (Contributed by NM, 22-Jun-2013.)
((φ ψ) → (χθ))       ((φ χ) → (ψθ))
 
Theoremcon3and 428 Variant of con3d 125 with importation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(φ → (ψχ))       ((φ ¬ χ) → ¬ ψ)
 
Theorempm2.01da 429 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
((φ ψ) → ¬ ψ)       (φ → ¬ ψ)
 
Theorempm2.18da 430 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
((φ ¬ ψ) → ψ)       (φψ)
 
Theorempm3.3 431 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)))
 
Theorempm3.31 432 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 433 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)))
 
Theorempm3.2 434 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.)
(φ → (ψ → (φ ψ)))
 
Theorempm3.21 435 Join antecedents with conjunction. Theorem *3.21 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.)
(φ → (ψ → (ψ φ)))
 
Theorempm3.22 436 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 437 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 438 Commutation of conjuncts in consequent. (Contributed by Jeff Hankins, 14-Aug-2009.)
(φ → (ψ χ))       (φ → (χ ψ))
 
Theoremancoms 439 Inference commuting conjunction in antecedent. (Contributed by NM, 21-Apr-1994.)
((φ ψ) → χ)       ((ψ φ) → χ)
 
Theoremancomsd 440 Deduction commuting conjunction in antecedent. (Contributed by NM, 12-Dec-2004.)
(φ → ((ψ χ) → θ))       (φ → ((χ ψ) → θ))
 
Theorempm3.2i 441 Infer conjunction of premises. (Contributed by NM, 5-Aug-1993.)
φ    &   ψ       (φ ψ)
 
Theorempm3.43i 442 Nested conjunction of antecedents. (Contributed by NM, 5-Aug-1993.)
((φψ) → ((φχ) → (φ → (ψ χ))))
 
Theoremsimpl 443 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.)
((φ ψ) → φ)
 
Theoremsimpli 444 Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)
(φ ψ)       φ
 
Theoremsimpld 445 Deduction eliminating a conjunct. A translation of natural deduction rule EL ( elimination left), see natded in set.mm. (Contributed by NM, 5-Aug-1993.)
(φ → (ψ χ))       (φψ)
 
Theoremsimplbi 446 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
(φ ↔ (ψ χ))       (φψ)
 
Theoremsimpr 447 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.)
((φ ψ) → ψ)
 
Theoremsimpri 448 Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)
(φ ψ)       ψ
 
Theoremsimprd 449 Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.) A translation of natural deduction rule ER ( elimination right), see natded in set.mm. (Proof shortened by Wolf Lammen, 3-Oct-2013.)
(φ → (ψ χ))       (φχ)
 
Theoremsimprbi 450 Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
(φ ↔ (ψ χ))       (φχ)
 
Theoremadantr 451 Inference adding a conjunct to the right of an antecedent. (Contributed by NM, 30-Aug-1993.)
(φψ)       ((φ χ) → ψ)
 
Theoremadantl 452 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 453 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 454 Deduction adding a conjunct to the right of an antecedent. (Contributed by NM, 4-May-1994.)
(φ → (ψχ))       (φ → ((ψ θ) → χ))
 
Theoremmpan9 455 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 456 A syllogism deduction with conjoined antecedents. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 6-Apr-2013.)
((φ ψ) → χ)    &   ((φ χ) → θ)       ((φ ψ) → θ)
 
Theoremsylan 457 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 458 A syllogism inference. (Contributed by NM, 18-May-1994.)
(φψ)    &   ((ψ χ) → θ)       ((φ χ) → θ)
 
Theoremsylanbr 459 A syllogism inference. (Contributed by NM, 18-May-1994.)
(ψφ)    &   ((ψ χ) → θ)       ((φ χ) → θ)
 
Theoremsylan2 460 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 461 A syllogism inference. (Contributed by NM, 21-Apr-1994.)
(φχ)    &   ((ψ χ) → θ)       ((ψ φ) → θ)
 
Theoremsylan2br 462 A syllogism inference. (Contributed by NM, 21-Apr-1994.)
(χφ)    &   ((ψ χ) → θ)       ((ψ φ) → θ)
 
Theoremsyl2an 463 A double syllogism inference. (Contributed by NM, 31-Jan-1997.)
(φψ)    &   (τχ)    &   ((ψ χ) → θ)       ((φ τ) → θ)
 
Theoremsyl2anr 464 A double syllogism inference. (Contributed by NM, 17-Sep-2013.)
(φψ)    &   (τχ)    &   ((ψ χ) → θ)       ((τ φ) → θ)
 
Theoremsyl2anb 465 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
(φψ)    &   (τχ)    &   ((ψ χ) → θ)       ((φ τ) → θ)
 
Theoremsyl2anbr 466 A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
(ψφ)    &   (χτ)    &   ((ψ χ) → θ)       ((φ τ) → θ)
 
Theoremsyland 467 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
(φ → (ψχ))    &   (φ → ((χ θ) → τ))       (φ → ((ψ θ) → τ))
 
Theoremsylan2d 468 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
(φ → (ψχ))    &   (φ → ((θ χ) → τ))       (φ → ((θ ψ) → τ))
 
Theoremsyl2and 469 A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
(φ → (ψχ))    &   (φ → (θτ))    &   (φ → ((χ τ) → η))       (φ → ((ψ θ) → η))
 
Theorembiimpa 470 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(φ → (ψχ))       ((φ ψ) → χ)
 
Theorembiimpar 471 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(φ → (ψχ))       ((φ χ) → ψ)
 
Theorembiimpac 472 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(φ → (ψχ))       ((ψ φ) → χ)
 
Theorembiimparc 473 Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(φ → (ψχ))       ((χ φ) → ψ)
 
Theoremianor 474 Negated conjunction in terms of disjunction (De Morgan's law). Theorem *4.51 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(¬ (φ ψ) ↔ (¬ φ ¬ ψ))
 
Theoremanor 475 Conjunction in terms of disjunction (De Morgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Nov-2012.)
((φ ψ) ↔ ¬ (¬ φ ¬ ψ))
 
Theoremioran 476 Negated disjunction in terms of conjunction (De Morgan's law). Compare Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(¬ (φ ψ) ↔ (¬ φ ¬ ψ))
 
Theorempm4.52 477 Theorem *4.52 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Nov-2012.)
((φ ¬ ψ) ↔ ¬ (¬ φ ψ))
 
Theorempm4.53 478 Theorem *4.53 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (φ ¬ ψ) ↔ (¬ φ ψ))
 
Theorempm4.54 479 Theorem *4.54 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Nov-2012.)
((¬ φ ψ) ↔ ¬ (φ ¬ ψ))
 
Theorempm4.55 480 Theorem *4.55 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (¬ φ ψ) ↔ (φ ¬ ψ))
 
Theorempm4.56 481 Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
((¬ φ ¬ ψ) ↔ ¬ (φ ψ))
 
Theoremoran 482 Disjunction in terms of conjunction (De Morgan's law). Compare Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)
((φ ψ) ↔ ¬ (¬ φ ¬ ψ))
 
Theorempm4.57 483 Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (¬ φ ¬ ψ) ↔ (φ ψ))
 
Theorempm3.1 484 Theorem *3.1 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)
((φ ψ) → ¬ (¬ φ ¬ ψ))
 
Theorempm3.11 485 Theorem *3.11 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)
(¬ (¬ φ ¬ ψ) → (φ ψ))
 
Theorempm3.12 486 Theorem *3.12 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)
((¬ φ ¬ ψ) (φ ψ))
 
Theorempm3.13 487 Theorem *3.13 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)
(¬ (φ ψ) → (¬ φ ¬ ψ))
 
Theorempm3.14 488 Theorem *3.14 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)
((¬ φ ¬ ψ) → ¬ (φ ψ))
 
Theoremiba 489 Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Mar-1994.)
(φ → (ψ ↔ (ψ φ)))
 
Theoremibar 490 Introduction of antecedent as conjunct. (Contributed by NM, 5-Dec-1995.)
(φ → (ψ ↔ (φ ψ)))
 
Theorembiantru 491 A wff is equivalent to its conjunction with truth. (Contributed by NM, 5-Aug-1993.)
φ       (ψ ↔ (ψ φ))
 
Theorembiantrur 492 A wff is equivalent to its conjunction with truth. (Contributed by NM, 3-Aug-1994.)
φ       (ψ ↔ (φ ψ))
 
Theorembiantrud 493 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 494 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)))
 
Theoremjaao 495 Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)
(φ → (ψχ))    &   (θ → (τχ))       ((φ θ) → ((ψ τ) → χ))
 
Theoremjaoa 496 Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)
(φ → (ψχ))    &   (θ → (τχ))       ((φ θ) → ((ψ τ) → χ))
 
Theorempm3.44 497 Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
(((ψφ) (χφ)) → ((ψ χ) → φ))
 
Theoremjao 498 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.)
((φψ) → ((χψ) → ((φ χ) → ψ)))
 
Theorempm1.2 499 Axiom *1.2 of [WhiteheadRussell] p. 96, which they call "Taut". (Contributed by NM, 3-Jan-2005.)
((φ φ) → φ)
 
Theoremoridm 500 Idempotent law for disjunction. Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 16-Apr-2011.) (Proof shortened by Wolf Lammen, 10-Mar-2013.)
((φ φ) ↔ φ)
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