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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | xchbinx 301 | Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.) |
Theorem | xchbinxr 302 | Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.) |
Theorem | imbi2i 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.) |
Theorem | bibi2i 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.) |
Theorem | bibi1i 305 | Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | bibi12i 306 | The equivalence of two equivalences. (Contributed by NM, 5-Aug-1993.) |
Theorem | imbi2d 307 | Deduction adding an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | imbi1d 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.) |
Theorem | bibi2d 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.) |
Theorem | bibi1d 310 | Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.) |
Theorem | imbi12d 311 | Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.) |
Theorem | bibi12d 312 | Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 5-Aug-1993.) |
Theorem | imbi1 313 | Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) |
Theorem | imbi2 314 | Theorem *4.85 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.) |
Theorem | imbi1i 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.) |
Theorem | imbi12i 316 | Join two logical equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.) |
Theorem | bibi1 317 | Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) |
Theorem | con2bi 318 | Contraposition. Theorem *4.12 of [WhiteheadRussell] p. 117. (Contributed by NM, 15-Apr-1995.) (Proof shortened by Wolf Lammen, 3-Jan-2013.) |
Theorem | con2bid 319 | A contraposition deduction. (Contributed by NM, 15-Apr-1995.) |
Theorem | con1bid 320 | A contraposition deduction. (Contributed by NM, 9-Oct-1999.) |
Theorem | con1bii 321 | A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.) |
Theorem | con2bii 322 | A contraposition inference. (Contributed by NM, 5-Aug-1993.) |
Theorem | con1b 323 | Contraposition. Bidirectional version of con1 120. (Contributed by NM, 5-Aug-1993.) |
Theorem | con2b 324 | Contraposition. Bidirectional version of con2 108. (Contributed by NM, 5-Aug-1993.) |
Theorem | biimt 325 | A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.) |
Theorem | pm5.5 326 | Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
Theorem | a1bi 327 | Inference rule introducing a theorem as an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.) |
Theorem | mt2bi 328 | A false consequent falsifies an antecedent. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.) |
Theorem | mtt 329 | Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Proof shortened by Wolf Lammen, 12-Nov-2012.) |
Theorem | pm5.501 330 | Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
Theorem | ibib 331 | Implication in terms of implication and biconditional. (Contributed by NM, 31-Mar-1994.) (Proof shortened by Wolf Lammen, 24-Jan-2013.) |
Theorem | ibibr 332 | Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.) |
Theorem | tbt 333 | A wff is equivalent to its equivalence with truth. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | nbn2 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.) |
Theorem | bibif 335 | Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.) |
Theorem | nbn 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.) |
Theorem | nbn3 337 | Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.) |
Theorem | pm5.21im 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.) |
Theorem | 2false 339 | Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.) |
Theorem | 2falsed 340 | Two falsehoods are equivalent (deduction rule). (Contributed by NM, 11-Oct-2013.) |
Theorem | pm5.21ni 341 | Two propositions implying a false one are equivalent. (Contributed by NM, 16-Feb-1996.) (Proof shortened by Wolf Lammen, 19-May-2013.) |
Theorem | pm5.21nii 342 | Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 21-May-1999.) |
Theorem | pm5.21ndd 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.) |
Theorem | bija 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.) |
Theorem | pm5.18 345 | 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.) |
Theorem | xor3 346 | Two ways to express "exclusive or." (Contributed by NM, 1-Jan-2006.) |
Theorem | nbbn 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.) |
Theorem | biass 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.) |
Theorem | pm5.19 349 | Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) |
Theorem | bi2.04 350 | Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 5-Aug-1993.) |
Theorem | pm5.4 351 | Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.) |
Theorem | imdi 352 | Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.) |
Theorem | pm5.41 353 | Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 12-Oct-2012.) |
Theorem | pm4.8 354 | Theorem *4.8 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm4.81 355 | Theorem *4.81 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) |
Theorem | imim21b 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.) |
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. | ||
Syntax | wo 357 | Extend wff definition to include disjunction ('or'). |
Syntax | wa 358 | Extend wff definition to include conjunction ('and'). |
Definition | df-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.) |
Definition | df-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.) |
Theorem | pm4.64 361 | Theorem *4.64 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.53 362 | Theorem *2.53 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.54 363 | Theorem *2.54 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
Theorem | ori 364 | Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) |
Theorem | orri 365 | Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) |
Theorem | ord 366 | Deduce implication from disjunction. (Contributed by NM, 18-May-1994.) |
Theorem | orrd 367 | Deduce implication from disjunction. (Contributed by NM, 27-Nov-1995.) |
Theorem | jaoi 368 | Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.) |
Theorem | jaod 369 | Deduction disjoining the antecedents of two implications. (Contributed by NM, 18-Aug-1994.) |
Theorem | mpjaod 370 | Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.) |
Theorem | orel1 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.) |
Theorem | orel2 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.) |
Theorem | olc 373 | Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.) |
Theorem | orc 374 | Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.) |
Theorem | pm1.4 375 | Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) |
Theorem | orcom 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.) |
Theorem | orcomd 377 | Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.) |
Theorem | orcoms 378 | Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.) |
Theorem | orci 379 | Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.) |
Theorem | olci 380 | Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.) |
Theorem | orcd 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.) |
Theorem | olcd 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.) |
Theorem | orcs 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.) |
Theorem | olcs 384 | Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) |
Theorem | pm2.07 385 | Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.45 386 | Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.46 387 | Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.47 388 | Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.48 389 | Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.49 390 | Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.67-2 391 | Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.67 392 | Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.25 393 | Theorem *2.25 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.) |
Theorem | biorf 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.) |
Theorem | biortn 395 | A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.) |
Theorem | biorfi 396 | A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.) |
Theorem | pm2.621 397 | Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.62 398 | Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.) |
Theorem | pm2.68 399 | Theorem *2.68 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
Theorem | dfor2 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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