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Theorem List for New Foundations Explorer - 301-400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxchbinx 301 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)

Theoremxchbinxr 302 Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.)

Theoremimbi2i 303 Introduce an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 6-Feb-2013.)

Theorembibi2i 304 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.)

Theorembibi1i 305 Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)

Theorembibi12i 306 The equivalence of two equivalences. (Contributed by NM, 5-Aug-1993.)

Theoremimbi2d 307 Deduction adding an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theoremimbi1d 308 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.)

Theorembibi2d 309 Deduction adding a biconditional to the left in an equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.)

Theorembibi1d 310 Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)

Theoremimbi12d 311 Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)

Theorembibi12d 312 Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 5-Aug-1993.)

Theoremimbi1 313 Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)

Theoremimbi2 314 Theorem *4.85 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)

Theoremimbi1i 315 Introduce a consequent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 17-Sep-2013.)

Theoremimbi12i 316 Join two logical equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)

Theorembibi1 317 Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)

Theoremcon2bi 318 Contraposition. Theorem *4.12 of [WhiteheadRussell] p. 117. (Contributed by NM, 15-Apr-1995.) (Proof shortened by Wolf Lammen, 3-Jan-2013.)

Theoremcon2bid 319 A contraposition deduction. (Contributed by NM, 15-Apr-1995.)

Theoremcon1bid 320 A contraposition deduction. (Contributed by NM, 9-Oct-1999.)

Theoremcon1bii 321 A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.)

Theoremcon2bii 322 A contraposition inference. (Contributed by NM, 5-Aug-1993.)

Theoremcon1b 323 Contraposition. Bidirectional version of con1 120. (Contributed by NM, 5-Aug-1993.)

Theoremcon2b 324 Contraposition. Bidirectional version of con2 108. (Contributed by NM, 5-Aug-1993.)

Theorembiimt 325 A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.)

Theorempm5.5 326 Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)

Theorema1bi 327 Inference rule introducing a theorem as an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)

Theoremmt2bi 328 A false consequent falsifies an antecedent. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)

Theoremmtt 329 Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Proof shortened by Wolf Lammen, 12-Nov-2012.)

Theorempm5.501 330 Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)

Theoremibib 331 Implication in terms of implication and biconditional. (Contributed by NM, 31-Mar-1994.) (Proof shortened by Wolf Lammen, 24-Jan-2013.)

Theoremibibr 332 Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.)

Theoremtbt 333 A wff is equivalent to its equivalence with truth. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Theoremnbn2 334 The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by Juha Arpiainen, 19-Jan-2006.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)

Theorembibif 335 Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.)

Theoremnbn 336 The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)

Theoremnbn3 337 Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.)

Theorempm5.21im 338 Two propositions are equivalent if they are both false. Closed form of 2false 339. Equivalent to a bi2 189-like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.)

Theorem2false 339 Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)

Theorem2falsed 340 Two falsehoods are equivalent (deduction rule). (Contributed by NM, 11-Oct-2013.)

Theorempm5.21ni 341 Two propositions implying a false one are equivalent. (Contributed by NM, 16-Feb-1996.) (Proof shortened by Wolf Lammen, 19-May-2013.)

Theorempm5.21nii 342 Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 21-May-1999.)

Theorempm5.21ndd 343 Eliminate an antecedent implied by each side of a biconditional, deduction version. (Contributed by Paul Chapman, 21-Nov-2012.) (Proof shortened by Wolf Lammen, 6-Oct-2013.)

Theorembija 344 Combine antecedents into a single bi-conditional. This inference, reminiscent of ja 153, is reversible: The hypotheses can be deduced from the conclusion alone (see pm5.1im 229 and pm5.21im 338). (Contributed by Wolf Lammen, 13-May-2013.)

Theorempm5.18 345 Theorem *5.18 of [WhiteheadRussell] p. 124. This theorem says that logical equivalence is the same as negated "exclusive-or." (Contributed by NM, 28-Jun-2002.) (Proof shortened by Andrew Salmon, 20-Jun-2011.) (Proof shortened by Wolf Lammen, 15-Oct-2013.)

Theoremxor3 346 Two ways to express "exclusive or." (Contributed by NM, 1-Jan-2006.)

Theoremnbbn 347 Move negation outside of biconditional. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by NM, 27-Jun-2002.) (Proof shortened by Wolf Lammen, 20-Sep-2013.)

Theorembiass 348 Associative law for the biconditional. An axiom of system DS in Vladimir Lifschitz, "On calculational proofs", Annals of Pure and Applied Logic, 113:207-224, 2002, http://www.cs.utexas.edu/users/ai-lab/pub-view.php?PubID=26805. Interestingly, this law was not included in Principia Mathematica but was apparently first noted by Jan Lukasiewicz circa 1923. (Contributed by NM, 8-Jan-2005.) (Proof shortened by Juha Arpiainen, 19-Jan-2006.) (Proof shortened by Wolf Lammen, 21-Sep-2013.)

Theorempm5.19 349 Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.)

Theorembi2.04 350 Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 5-Aug-1993.)

Theorempm5.4 351 Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)

Theoremimdi 352 Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)

Theorempm5.41 353 Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 12-Oct-2012.)

Theorempm4.8 354 Theorem *4.8 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)

Theorempm4.81 355 Theorem *4.81 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)

Theoremimim21b 356 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.)

1.2.6  Logical disjunction and conjunction

Here we define disjunction (logical 'or') (df-or 359) and conjunction (logical 'and') (df-an 360). We also define various rules for simplifying and applying them, e.g., olc 373, orc 374, and orcom 376.

Syntaxwo 357 Extend wff definition to include disjunction ('or').

Syntaxwa 358 Extend wff definition to include conjunction ('and').

Definitiondf-or 359 Define disjunction (logical 'or'). Definition of [Margaris] p. 49. When the left operand, right operand, or both are true, the result is true; when both sides are false, the result is false. For example, it is true that (2 = 3 4 = 4) (see ex-or in set.mm). After we define the constant true (df-tru 1319) and the constant false (df-fal 1320), we will be able to prove these truth table values: (truortru 1340), (truorfal 1341), (falortru 1342), and (falorfal 1343).

This is our first use of the biconditional connective in a definition; we use the biconditional connective in place of the traditional "<=def=>", which means the same thing, except that we can manipulate the biconditional connective directly in proofs rather than having to rely on an informal definition substitution rule. Note that if we mechanically substitute for , we end up with an instance of previously proved theorem biid 227. This is the justification for the definition, along with the fact that it introduces a new symbol . Contrast with (df-an 360), (wi 4), (df-nan 1288), and (df-xor 1305) . (Contributed by NM, 5-Aug-1993.)

Definitiondf-an 360 Define conjunction (logical 'and'). Definition of [Margaris] p. 49. When both the left and right operand are true, the result is true; when either is false, the result is false. For example, it is true that (2 = 2 3 = 3). After we define the constant true (df-tru 1319) and the constant false (df-fal 1320), we will be able to prove these truth table values: (truantru 1336), (truanfal 1337), (falantru 1338), and (falanfal 1339).

Contrast with (df-or 359), (wi 4), (df-nan 1288), and (df-xor 1305) . (Contributed by NM, 5-Aug-1993.)

Theorempm4.64 361 Theorem *4.64 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)

Theorempm2.53 362 Theorem *2.53 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)

Theorempm2.54 363 Theorem *2.54 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)

Theoremori 364 Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.)

Theoremorri 365 Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.)

Theoremord 366 Deduce implication from disjunction. (Contributed by NM, 18-May-1994.)

Theoremorrd 367 Deduce implication from disjunction. (Contributed by NM, 27-Nov-1995.)

Theoremjaoi 368 Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.)

Theoremjaod 369 Deduction disjoining the antecedents of two implications. (Contributed by NM, 18-Aug-1994.)

Theoremmpjaod 370 Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremorel1 371 Elimination of disjunction by denial of a disjunct. Theorem *2.55 of [WhiteheadRussell] p. 107. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Wolf Lammen, 21-Jul-2012.)

Theoremorel2 372 Elimination of disjunction by denial of a disjunct. Theorem *2.56 of [WhiteheadRussell] p. 107. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Wolf Lammen, 5-Apr-2013.)

Theoremolc 373 Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.)

Theoremorc 374 Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.)

Theorempm1.4 375 Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)

Theoremorcom 376 Commutative law for disjunction. Theorem *4.31 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 15-Nov-2012.)

Theoremorcomd 377 Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)

Theoremorcoms 378 Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)

Theoremorci 379 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)

Theoremolci 380 Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)

Theoremorcd 381 Deduction introducing a disjunct. A translation of natural deduction rule IR ( insertion right), see natded in set.mm. (Contributed by NM, 20-Sep-2007.)

Theoremolcd 382 Deduction introducing a disjunct. A translation of natural deduction rule IL ( insertion left), see natded in set.mm. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)

Theoremorcs 383 Deduction eliminating disjunct. Notational convention: We sometimes suffix with "s" the label of an inference that manipulates an antecedent, leaving the consequent unchanged. The "s" means that the inference eliminates the need for a syllogism (syl 15) -type inference in a proof. (Contributed by NM, 21-Jun-1994.)

Theoremolcs 384 Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)

Theorempm2.07 385 Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)

Theorempm2.45 386 Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)

Theorempm2.46 387 Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)

Theorempm2.47 388 Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)

Theorempm2.48 389 Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)

Theorempm2.49 390 Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)

Theorempm2.67-2 391 Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)

Theorempm2.67 392 Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)

Theorempm2.25 393 Theorem *2.25 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)

Theorembiorf 394 A wff is equivalent to its disjunction with falsehood. Theorem *4.74 of [WhiteheadRussell] p. 121. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)

Theorembiortn 395 A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)

Theorembiorfi 396 A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.)

Theorempm2.621 397 Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)

Theorempm2.62 398 Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.)

Theorempm2.68 399 Theorem *2.68 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.)

Theoremdfor2 400 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.)

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