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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 3bitr4rd 301 | Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.) |
Theorem | 3bitr3g 302 | More general version of 3bitr3i 290. Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.) |
Theorem | 3bitr4g 303 | More general version of 3bitr4i 292. Useful for converting definitions in a formula. (Contributed by NM, 11-May-1993.) |
Theorem | notnotb 304 | Double negation. Theorem *4.13 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-1993.) |
Theorem | notnotdOLD 305 | Obsolete proof of notnotd 138 as of 27-Mar-2021. (Contributed by Jarvin Udandy, 2-Sep-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | con34b 306 | A biconditional form of contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116. (Contributed by NM, 11-May-1993.) |
Theorem | con4bid 307 | A contraposition deduction. (Contributed by NM, 21-May-1994.) |
Theorem | notbid 308 | Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994.) |
Theorem | notbi 309 | Contraposition. Theorem *4.11 of [WhiteheadRussell] p. 117. (Contributed by NM, 21-May-1994.) (Proof shortened by Wolf Lammen, 12-Jun-2013.) |
Theorem | notbii 310 | Negate both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.) |
Theorem | con4bii 311 | A contraposition inference. (Contributed by NM, 21-May-1994.) |
Theorem | mtbi 312 | An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Oct-2012.) |
Theorem | mtbir 313 | An inference from a biconditional, related to modus tollens. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Wolf Lammen, 14-Oct-2012.) |
Theorem | mtbid 314 | A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 26-Nov-1995.) |
Theorem | mtbird 315 | A deduction from a biconditional, similar to modus tollens. (Contributed by NM, 10-May-1994.) |
Theorem | mtbii 316 | An inference from a biconditional, similar to modus tollens. (Contributed by NM, 27-Nov-1995.) |
Theorem | mtbiri 317 | An inference from a biconditional, similar to modus tollens. (Contributed by NM, 24-Aug-1995.) |
Theorem | sylnib 318 | A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.) |
Theorem | sylnibr 319 | A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.) |
Theorem | sylnbi 320 | A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by Wolf Lammen, 16-Dec-2013.) |
Theorem | sylnbir 321 | A mixed syllogism inference from a biconditional and an implication. (Contributed by Wolf Lammen, 16-Dec-2013.) |
Theorem | xchnxbi 322 | Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.) |
Theorem | xchnxbir 323 | Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.) |
Theorem | xchbinx 324 | Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.) |
Theorem | xchbinxr 325 | Replacement of a subexpression by an equivalent one. (Contributed by Wolf Lammen, 27-Sep-2014.) |
Theorem | imbi2i 326 | Introduce an antecedent to both sides of a logical equivalence. This and the next three rules are useful for building up wff's around a definition, in order to make use of the definition. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 6-Feb-2013.) |
Theorem | bibi2i 327 | Inference adding a biconditional to the left in an equivalence. (Contributed by NM, 26-May-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 16-May-2013.) |
Theorem | bibi1i 328 | Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 26-May-1993.) |
Theorem | bibi12i 329 | The equivalence of two equivalences. (Contributed by NM, 26-May-1993.) |
Theorem | imbi2d 330 | Deduction adding an antecedent to both sides of a logical equivalence. (Contributed by NM, 11-May-1993.) |
Theorem | imbi1d 331 | Deduction adding a consequent to both sides of a logical equivalence. (Contributed by NM, 11-May-1993.) (Proof shortened by Wolf Lammen, 17-Sep-2013.) |
Theorem | bibi2d 332 | Deduction adding a biconditional to the left in an equivalence. (Contributed by NM, 11-May-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.) |
Theorem | bibi1d 333 | Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 11-May-1993.) |
Theorem | imbi12d 334 | Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 16-May-1993.) |
Theorem | bibi12d 335 | Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 26-May-1993.) |
Theorem | imbi12 336 | Closed form of imbi12i 340. Was automatically derived from its "Virtual Deduction" version and Metamath's "minimize" command. (Contributed by Alan Sare, 18-Mar-2012.) |
Theorem | imbi1 337 | Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) |
Theorem | imbi2 338 | Theorem *4.85 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.) |
Theorem | imbi1i 339 | Introduce a consequent to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 17-Sep-2013.) |
Theorem | imbi12i 340 | Join two logical equivalences to form equivalence of implications. (Contributed by NM, 1-Aug-1993.) |
Theorem | bibi1 341 | Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) |
Theorem | bitr3 342 | Closed nested implication form of bitr3i 266. Derived automatically from bitr3VD 39084. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | con2bi 343 | Contraposition. Theorem *4.12 of [WhiteheadRussell] p. 117. (Contributed by NM, 15-Apr-1995.) (Proof shortened by Wolf Lammen, 3-Jan-2013.) |
Theorem | con2bid 344 | A contraposition deduction. (Contributed by NM, 15-Apr-1995.) |
Theorem | con1bid 345 | A contraposition deduction. (Contributed by NM, 9-Oct-1999.) |
Theorem | con1bii 346 | A contraposition inference. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.) |
Theorem | con2bii 347 | A contraposition inference. (Contributed by NM, 12-Mar-1993.) |
Theorem | con1b 348 | Contraposition. Bidirectional version of con1 143. (Contributed by NM, 3-Jan-1993.) |
Theorem | con2b 349 | Contraposition. Bidirectional version of con2 130. (Contributed by NM, 12-Mar-1993.) |
Theorem | biimt 350 | A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.) |
Theorem | pm5.5 351 | Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
Theorem | a1bi 352 | Inference rule introducing a theorem as an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.) |
Theorem | mt2bi 353 | A false consequent falsifies an antecedent. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.) |
Theorem | mtt 354 | Modus-tollens-like theorem. (Contributed by NM, 7-Apr-2001.) (Proof shortened by Wolf Lammen, 12-Nov-2012.) |
Theorem | imnot 355 | If a proposition is false, then implying it is equivalent to being false. One of four theorems that can be used to simplify an implication , the other ones being ax-1 6 (true consequent), pm2.21 120 (false antecedent), pm5.5 351 (true antecedent). (Contributed by Mario Carneiro, 26-Apr-2019.) (Proof shortened by Wolf Lammen, 26-May-2019.) |
Theorem | pm5.501 356 | Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
Theorem | ibib 357 | Implication in terms of implication and biconditional. (Contributed by NM, 31-Mar-1994.) (Proof shortened by Wolf Lammen, 24-Jan-2013.) |
Theorem | ibibr 358 | Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.) |
Theorem | tbt 359 | A wff is equivalent to its equivalence with a truth. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | nbn2 360 | 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 361 | Transfer negation via an equivalence. (Contributed by NM, 3-Oct-2007.) (Proof shortened by Wolf Lammen, 28-Jan-2013.) |
Theorem | nbn 362 | The negation of a wff is equivalent to the wff's equivalence to falsehood. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) |
Theorem | nbn3 363 | Transfer falsehood via equivalence. (Contributed by NM, 11-Sep-2006.) |
Theorem | pm5.21im 364 | Two propositions are equivalent if they are both false. Closed form of 2false 365. Equivalent to a biimpr 210-like version of the xor-connective. (Contributed by Wolf Lammen, 13-May-2013.) |
Theorem | 2false 365 | Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.) |
Theorem | 2falsed 366 | Two falsehoods are equivalent (deduction rule). (Contributed by NM, 11-Oct-2013.) |
Theorem | pm5.21ni 367 | 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 368 | Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 21-May-1999.) |
Theorem | pm5.21ndd 369 | 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 370 | Combine antecedents into a single biconditional. This inference, reminiscent of ja 173, is reversible: The hypotheses can be deduced from the conclusion alone (see pm5.1im 253 and pm5.21im 364). (Contributed by Wolf Lammen, 13-May-2013.) |
Theorem | pm5.18 371 | 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 372 | Two ways to express "exclusive or." (Contributed by NM, 1-Jan-2006.) |
Theorem | nbbn 373 | 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 374 | 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 | pm5.19 375 | Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) |
Theorem | bi2.04 376 | Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 11-May-1993.) |
Theorem | pm5.4 377 | Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.) |
Theorem | imdi 378 | Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.) |
Theorem | pm5.41 379 | Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 12-Oct-2012.) |
Theorem | pm4.8 380 | Theorem *4.8 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm4.81 381 | Theorem *4.81 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) |
Theorem | imim21b 382 | 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 385) and conjunction (logical 'and') (df-an 386). We also define various rules for simplifying and applying them, e.g., olc 399, orc 400, and orcom 402. | ||
Syntax | wo 383 | Extend wff definition to include disjunction ('or'). |
Syntax | wa 384 | Extend wff definition to include conjunction ('and'). |
Definition | df-or 385 |
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 (ex-or 27278). After we define the constant
true (df-tru 1486) and the constant false (df-fal 1489), we
will be able to prove these truth table values:
(truortru 1510),
(truorfal 1511),
(falortru 1512), and
(falorfal 1513).
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 251. This is the justification for the definition, along with the fact that it introduces a new symbol . Contrast with (df-an 386), (wi 4), (df-nan 1448), and (df-xor 1465) . (Contributed by NM, 27-Dec-1992.) |
Definition | df-an 386 |
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
. After we define the
constant true
(df-tru 1486) and the constant false (df-fal 1489), we will be able
to prove these truth table values:
(truantru 1506),
(truanfal 1507),
(falantru 1508), and
(falanfal 1509).
Contrast with (df-or 385), (wi 4), (df-nan 1448), and (df-xor 1465) . (Contributed by NM, 5-Jan-1993.) |
Theorem | pm4.64 387 | Theorem *4.64 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.53 388 | Theorem *2.53 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
Theorem | pm2.54 389 | Theorem *2.54 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
Theorem | ori 390 | Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) |
Theorem | orri 391 | Infer disjunction from implication. (Contributed by NM, 11-Jun-1994.) |
Theorem | ord 392 | Deduce implication from disjunction. (Contributed by NM, 18-May-1994.) |
Theorem | orrd 393 | Deduce disjunction from implication. (Contributed by NM, 27-Nov-1995.) |
Theorem | jaoi 394 | Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.) |
Theorem | jaod 395 | Deduction disjoining the antecedents of two implications. (Contributed by NM, 18-Aug-1994.) |
Theorem | mpjaod 396 | Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.) |
Theorem | orel1 397 | 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 398 | 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 399 | Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.) |
Theorem | orc 400 | Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.) |
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