HomeHome Intuitionistic Logic Explorer
Theorem List (p. 14 of 131)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 1301-1400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmp3and 1301 A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  ( ( ps  /\  ch  /\ 
 th )  ->  ta )
 )   =>    |-  ( ph  ->  ta )
 
Theoremmp3an12i 1302 mp3an 1298 with antecedents in standard conjunction form and with one hypothesis an implication. (Contributed by Alan Sare, 28-Aug-2016.)
 |-  ph   &    |- 
 ps   &    |-  ( ch  ->  th )   &    |-  (
 ( ph  /\  ps  /\  th )  ->  ta )   =>    |-  ( ch  ->  ta )
 
Theoremmp3an2i 1303 mp3an 1298 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  ( ps  ->  th )   &    |-  ( ( ph  /\ 
 ch  /\  th )  ->  ta )   =>    |-  ( ps  ->  ta )
 
Theoremmp3an3an 1304 mp3an 1298 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  ( th  ->  ta )   &    |-  ( ( ph  /\ 
 ch  /\  ta )  ->  et )   =>    |-  ( ( ps  /\  th )  ->  et )
 
Theoremmp3an2ani 1305 An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  (
 ( ps  /\  th )  ->  ta )   &    |-  ( ( ph  /\ 
 ch  /\  ta )  ->  et )   =>    |-  ( ( ps  /\  th )  ->  et )
 
Theorembiimp3a 1306 Infer implication from a logical equivalence. Similar to biimpa 292. (Contributed by NM, 4-Sep-2005.)
 |-  ( ( ph  /\  ps )  ->  ( ch  <->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembiimp3ar 1307 Infer implication from a logical equivalence. Similar to biimpar 293. (Contributed by NM, 2-Jan-2009.)
 |-  ( ( ph  /\  ps )  ->  ( ch  <->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  th )  ->  ch )
 
Theorem3anandis 1308 Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ph  /\  ch )  /\  ( ph  /\  th )
 )  ->  ta )   =>    |-  (
 ( ph  /\  ( ps 
 /\  ch  /\  th )
 )  ->  ta )
 
Theorem3anandirs 1309 Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.) (Revised by NM, 18-Apr-2007.)
 |-  ( ( ( ph  /\ 
 th )  /\  ( ps  /\  th )  /\  ( ch  /\  th )
 )  ->  ta )   =>    |-  (
 ( ( ph  /\  ps  /\ 
 ch )  /\  th )  ->  ta )
 
Theoremecased 1310 Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  ( ps  \/  ch ) )   =>    |-  ( ph  ->  ps )
 
Theoremecase23d 1311 Variation of ecased 1310 with three disjuncts instead of two. (Contributed by NM, 22-Apr-1994.) (Revised by Jim Kingdon, 9-Dec-2017.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  -.  th )   &    |-  ( ph  ->  ( ps  \/  ch  \/  th ) )   =>    |-  ( ph  ->  ps )
 
1.2.12  True and false constants
 
1.2.12.1  Universal quantifier for use by df-tru

Even though it isn't ordinarily part of propositional calculus, the universal quantifier  A. is introduced here so that the soundness of definition df-tru 1317 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in axiom ax-5 1406 in the predicate calculus section below. For those who want propositional calculus to be self-contained i.e. to use wff variables only, the alternate definition dftru2 1322 may be adopted and this subsection moved down to the start of the subsection with wex 1451 below. However, the use of dftru2 1322 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

 
Syntaxwal 1312 Extend wff definition to include the universal quantifier ('for all').  A. x ph is read " ph (phi) is true for all  x." Typically, in its final application 
ph would be replaced with a wff containing a (free) occurrence of the variable  x, for example  x  =  y. In a universe with a finite number of objects, "for all" is equivalent to a big conjunction (AND) with one wff for each possible case of  x. When the universe is infinite (as with set theory), such a propositional-calculus equivalent is not possible because an infinitely long formula has no meaning, but conceptually the idea is the same.
 wff  A. x ph
 
1.2.12.2  Equality predicate for use by df-tru

Even though it isn't ordinarily part of propositional calculus, the equality predicate  = is introduced here so that the soundness of definition df-tru 1317 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in axiom ax-8 1465 in the predicate calculus section below. For those who want propositional calculus to be self-contained i.e. to use wff variables only, the alternate definition dftru2 1322 may be adopted and this subsection moved down to just above weq 1462 below. However, the use of dftru2 1322 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

 
Syntaxcv 1313 This syntax construction states that a variable  x, which has been declared to be a setvar variable by $f statement vx, is also a class expression. This can be justified informally as follows. We know that the class builder  { y  |  y  e.  x } is a class by cab 2101. Since (when  y is distinct from  x) we have  x  =  { y  |  y  e.  x } by cvjust 2110, we can argue that the syntax " class  x " can be viewed as an abbreviation for "
class  { y  |  y  e.  x }". See the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class."

While it is tempting and perhaps occasionally useful to view cv 1313 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1313 is intrinsically no different from any other class-building syntax such as cab 2101, cun 3037, or c0 3331.

For a general discussion of the theory of classes and the role of cv 1313, see https://us.metamath.org/mpeuni/mmset.html#class 1313.

(The description above applies to set theory, not predicate calculus. The purpose of introducing  class  x here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1462 of predicate calculus from the wceq 1314 of set theory, so that we don't overload the  = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers.)

 class  x
 
Syntaxwceq 1314 Extend wff definition to include class equality.

For a general discussion of the theory of classes, see https://us.metamath.org/mpeuni/mmset.html#class.

(The purpose of introducing 
wff  A  =  B here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1462 of predicate calculus in terms of the wceq 1314 of set theory, so that we don't "overload" the  = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. For example, some parsers - although not the Metamath program - stumble on the fact that the  = in  x  =  y could be the  = of either weq 1462 or wceq 1314, although mathematically it makes no difference. The class variables  A and  B are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-cleq 2108 for more information on the set theory usage of wceq 1314.)

 wff  A  =  B
 
1.2.12.3  Define the true and false constants
 
Syntaxwtru 1315 T. is a wff.
 wff T.
 
Theoremtrujust 1316 Soundness justification theorem for df-tru 1317. (Contributed by Mario Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.)
 |-  ( ( A. x  x  =  x  ->  A. x  x  =  x )  <->  ( A. y  y  =  y  ->  A. y  y  =  y ) )
 
Definitiondf-tru 1317 Definition of the truth value "true", or "verum", denoted by T.. This is a tautology, as proved by tru 1318. In this definition, an instance of id 19 is used as the definiens, although any tautology, such as an axiom, can be used in its place. This particular id 19 instance was chosen so this definition can be checked by the same algorithm that is used for predicate calculus. This definition should be referenced directly only by tru 1318, and other proofs should depend on tru 1318 (directly or indirectly) instead of this definition, since there are many alternate ways to define T.. (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by NM, 11-Jul-2019.) (New usage is discouraged.)
 |-  ( T.  <->  ( A. x  x  =  x  ->  A. x  x  =  x ) )
 
Theoremtru 1318 The truth value T. is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
 |- T.
 
Syntaxwfal 1319 F. is a wff.
 wff F.
 
Definitiondf-fal 1320 Definition of the truth value "false", or "falsum", denoted by F.. See also df-tru 1317. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( F.  <->  -. T.  )
 
Theoremfal 1321 The truth value F. is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.)
 |- 
 -. F.
 
Theoremdftru2 1322 An alternate definition of "true". (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) (New usage is discouraged.)
 |-  ( T.  <->  ( ph  ->  ph ) )
 
Theoremmptru 1323 Eliminate T. as an antecedent. A proposition implied by T. is true. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  ( T.  ->  ph )   =>    |-  ph
 
Theoremtbtru 1324 A proposition is equivalent to itself being equivalent to T.. (Contributed by Anthony Hart, 14-Aug-2011.)
 |-  ( ph  <->  ( ph  <-> T.  ) )
 
Theoremnbfal 1325 The negation of a proposition is equivalent to itself being equivalent to F.. (Contributed by Anthony Hart, 14-Aug-2011.)
 |-  ( -.  ph  <->  ( ph  <-> F.  ) )
 
Theorembitru 1326 A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
 |-  ph   =>    |-  ( ph  <-> T.  )
 
Theorembifal 1327 A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
 |- 
 -.  ph   =>    |-  ( ph  <-> F.  )
 
Theoremfalim 1328 The truth value F. implies anything. Also called the principle of explosion, or "ex falso quodlibet". (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
 |-  ( F.  ->  ph )
 
Theoremfalimd 1329 The truth value F. implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ( ph  /\ F.  )  ->  ps )
 
Theorema1tru 1330 Anything implies T.. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
 |-  ( ph  -> T.  )
 
Theoremtruan 1331 True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
 |-  ( ( T.  /\  ph )  <->  ph )
 
Theoremdfnot 1332 Given falsum, we can define the negation of a wff  ph as the statement that a contradiction follows from assuming  ph. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 21-Jul-2019.)
 |-  ( -.  ph  <->  ( ph  -> F.  ) )
 
Theoreminegd 1333 Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ( ph  /\  ps )  -> F.  )   =>    |-  ( ph  ->  -. 
 ps )
 
Theorempm2.21fal 1334 If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  -.  ps )   =>    |-  ( ph  -> F.  )
 
Theorempclem6 1335 Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof rewritten by Jim Kingdon, 4-May-2018.)
 |-  ( ( ph  <->  ( ps  /\  -.  ph ) )  ->  -.  ps )
 
1.2.13  Logical 'xor'
 
Syntaxwxo 1336 Extend wff definition to include exclusive disjunction ('xor').
 wff  ( ph  \/_  ps )
 
Definitiondf-xor 1337 Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. Contrast with  /\ (wa 103),  \/ (wo 680), and  -> (wi 4) . (Contributed by FL, 22-Nov-2010.) (Modified by Jim Kingdon, 1-Mar-2018.)
 |-  ( ( ph  \/_  ps ) 
 <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\  ps ) ) )
 
Theoremxoranor 1338 One way of defining exclusive or. Equivalent to df-xor 1337. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.)
 |-  ( ( ph  \/_  ps ) 
 <->  ( ( ph  \/  ps )  /\  ( -.  ph  \/  -.  ps )
 ) )
 
Theoremexcxor 1339 This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.) (Proof rewritten by Jim Kingdon, 5-May-2018.)
 |-  ( ( ph  \/_  ps ) 
 <->  ( ( ph  /\  -.  ps )  \/  ( -.  ph  /\  ps ) ) )
 
Theoremxoror 1340 XOR implies OR. (Contributed by BJ, 19-Apr-2019.)
 |-  ( ( ph  \/_  ps )  ->  ( ph  \/  ps ) )
 
Theoremxorbi2d 1341 Deduction joining an equivalence and a left operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( th  \/_  ps ) 
 <->  ( th  \/_  ch ) ) )
 
Theoremxorbi1d 1342 Deduction joining an equivalence and a right operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( ps  \/_  th )  <->  ( ch  \/_  th )
 ) )
 
Theoremxorbi12d 1343 Deduction joining two equivalences to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ( ps  \/_  th )  <->  ( ch  \/_  ta )
 ) )
 
Theoremxorbi12i 1344 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  \/_  ch ) 
 <->  ( ps  \/_  th )
 )
 
Theoremxorbin 1345 A consequence of exclusive or. In classical logic the converse also holds. (Contributed by Jim Kingdon, 8-Mar-2018.)
 |-  ( ( ph  \/_  ps )  ->  ( ph  <->  -.  ps ) )
 
Theorempm5.18im 1346 One direction of pm5.18dc 851, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.)
 |-  ( ( ph  <->  ps )  ->  -.  ( ph 
 <->  -.  ps ) )
 
Theoremxornbi 1347 A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1352. (Contributed by Jim Kingdon, 10-Mar-2018.)
 |-  ( ( ph  \/_  ps )  ->  -.  ( ph  <->  ps ) )
 
Theoremxor3dc 1348 Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 12-Apr-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( -.  ( ph 
 <->  ps )  <->  ( ph  <->  -.  ps ) ) ) )
 
Theoremxorcom 1349  \/_ is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.)
 |-  ( ( ph  \/_  ps ) 
 <->  ( ps  \/_  ph )
 )
 
Theorempm5.15dc 1350 A decidable proposition is equivalent to a decidable proposition or its negation. Based on theorem *5.15 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( ( ph  <->  ps )  \/  ( ph  <->  -.  ps ) ) ) )
 
Theoremxor2dc 1351 Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 17-Apr-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( -.  ( ph 
 <->  ps )  <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\  ps )
 ) ) ) )
 
Theoremxornbidc 1352 Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( ( ph  \/_ 
 ps )  <->  -.  ( ph  <->  ps ) ) ) )
 
Theoremxordc 1353 Two ways to express "exclusive or" between decidable propositions. Theorem *5.22 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( -.  ( ph 
 <->  ps )  <->  ( ( ph  /\ 
 -.  ps )  \/  ( ps  /\  -.  ph )
 ) ) ) )
 
Theoremxordc1 1354 Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.)
 |-  ( ( ph  \/_  ps )  -> DECID  ph )
 
Theoremnbbndc 1355 Move negation outside of biconditional, for decidable propositions. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( ( -.  ph 
 <->  ps )  <->  -.  ( ph  <->  ps ) ) ) )
 
Theorembiassdc 1356 Associative law for the biconditional, for decidable propositions.

The classical version (without the decidability conditions) is 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, and, interestingly, was not included in Principia Mathematica but was apparently first noted by Jan Lukasiewicz circa 1923. (Contributed by Jim Kingdon, 4-May-2018.)

 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  (DECID 
 ch  ->  ( ( (
 ph 
 <->  ps )  <->  ch )  <->  ( ph  <->  ( ps  <->  ch ) ) ) ) ) )
 
Theorembilukdc 1357 Lukasiewicz's shortest axiom for equivalential calculus (but modified to require decidable propositions). Storrs McCall, ed., Polish Logic 1920-1939 (Oxford, 1967), p. 96. (Contributed by Jim Kingdon, 5-May-2018.)
 |-  ( ( (DECID  ph  /\ DECID  ps )  /\ DECID  ch )  ->  ( ( ph 
 <->  ps )  <->  ( ( ch  <->  ps )  <->  ( ph  <->  ch ) ) ) )
 
Theoremdfbi3dc 1358 An alternate definition of the biconditional for decidable propositions. Theorem *5.23 of [WhiteheadRussell] p. 124, but with decidability conditions. (Contributed by Jim Kingdon, 5-May-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( ( ph  <->  ps ) 
 <->  ( ( ph  /\  ps )  \/  ( -.  ph  /\ 
 -.  ps ) ) ) ) )
 
Theorempm5.24dc 1359 Theorem *5.24 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( -.  (
 ( ph  /\  ps )  \/  ( -.  ph  /\  -.  ps ) )  <->  ( ( ph  /\ 
 -.  ps )  \/  ( ps  /\  -.  ph )
 ) ) ) )
 
Theoremxordidc 1360 Conjunction distributes over exclusive-or, for decidable propositions. This is one way to interpret the distributive law of multiplication over addition in modulo 2 arithmetic. (Contributed by Jim Kingdon, 14-Jul-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  (DECID 
 ch  ->  ( ( ph  /\  ( ps  \/_  ch ) )  <->  ( ( ph  /\ 
 ps )  \/_  ( ph  /\  ch ) ) ) ) ) )
 
Theoremanxordi 1361 Conjunction distributes over exclusive-or. (Contributed by Mario Carneiro and Jim Kingdon, 7-Oct-2018.)
 |-  ( ( ph  /\  ( ps  \/_  ch ) )  <-> 
 ( ( ph  /\  ps )  \/_  ( ph  /\  ch ) ) )
 
1.2.14  Truth tables: Operations on true and false constants

For classical logic, truth tables can be used to define propositional logic operations, by showing the results of those operations for all possible combinations of true (T.) and false (F.).

Although the intuitionistic logic connectives are not as simply defined, T. and F. do play similar roles as in classical logic and most theorems from classical logic continue to hold.

Here we show that our definitions and axioms produce equivalent results for T. and F. as we would get from truth tables for  /\ (conjunction aka logical 'and') wa 103,  \/ (disjunction aka logical inclusive 'or') wo 680,  -> (implies) wi 4,  -. (not) wn 3,  <-> (logical equivalence) df-bi 116, and  \/_ (exclusive or) df-xor 1337.

 
Theoremtruantru 1362 A  /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( ( T.  /\ T.  )  <-> T.  )
 
Theoremtruanfal 1363 A  /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( ( T.  /\ F.  )  <-> F.  )
 
Theoremfalantru 1364 A  /\ identity. (Contributed by David A. Wheeler, 23-Feb-2018.)
 |-  ( ( F.  /\ T.  )  <-> F.  )
 
Theoremfalanfal 1365 A  /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( ( F.  /\ F.  )  <-> F.  )
 
Theoremtruortru 1366 A  \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( T.  \/ T.  )  <-> T.  )
 
Theoremtruorfal 1367 A  \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( ( T.  \/ F.  )  <-> T.  )
 
Theoremfalortru 1368 A  \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( ( F.  \/ T.  )  <-> T.  )
 
Theoremfalorfal 1369 A  \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( F.  \/ F.  )  <-> F.  )
 
Theoremtruimtru 1370 A  -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( ( T.  -> T.  )  <-> T.  )
 
Theoremtruimfal 1371 A  -> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( T.  -> F.  )  <-> F.  )
 
Theoremfalimtru 1372 A  -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( ( F.  -> T.  )  <-> T.  )
 
Theoremfalimfal 1373 A  -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( ( F.  -> F.  )  <-> T.  )
 
Theoremnottru 1374 A  -. identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( -. T.  <-> F.  )
 
Theoremnotfal 1375 A  -. identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( -. F.  <-> T.  )
 
Theoremtrubitru 1376 A  <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( T.  <-> T.  )  <-> T.  )
 
Theoremtrubifal 1377 A  <-> identity. (Contributed by David A. Wheeler, 23-Feb-2018.)
 |-  ( ( T.  <-> F.  )  <-> F.  )
 
Theoremfalbitru 1378 A  <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( F.  <-> T.  )  <-> F.  )
 
Theoremfalbifal 1379 A  <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( F.  <-> F.  )  <-> T.  )
 
Theoremtruxortru 1380 A  \/_ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
 |-  ( ( T.  \/_ T.  )  <-> F.  )
 
Theoremtruxorfal 1381 A  \/_ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
 |-  ( ( T.  \/_ F.  )  <-> T.  )
 
Theoremfalxortru 1382 A  \/_ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
 |-  ( ( F.  \/_ T.  )  <-> T.  )
 
Theoremfalxorfal 1383 A  \/_ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
 |-  ( ( F.  \/_ F.  )  <-> F.  )
 
1.2.15  Stoic logic indemonstrables (Chrysippus of Soli)

The Greek Stoics developed a system of logic. The Stoic Chrysippus, in particular, was often considered one of the greatest logicians of antiquity. Stoic logic is different from Aristotle's system, since it focuses on propositional logic, though later thinkers did combine the systems of the Stoics with Aristotle. Jan Lukasiewicz reports, "For anybody familiar with mathematical logic it is self-evident that the Stoic dialectic is the ancient form of modern propositional logic" ( On the history of the logic of proposition by Jan Lukasiewicz (1934), translated in: Selected Works - Edited by Ludwik Borkowski - Amsterdam, North-Holland, 1970 pp. 197-217, referenced in "History of Logic" https://www.historyoflogic.com/logic-stoics.htm). For more about Aristotle's system, see barbara and related theorems.

A key part of the Stoic logic system is a set of five "indemonstrables" assigned to Chrysippus of Soli by Diogenes Laertius, though in general it is difficult to assign specific ideas to specific thinkers. The indemonstrables are described in, for example, [Lopez-Astorga] p. 11 , [Sanford] p. 39, and [Hitchcock] p. 5. These indemonstrables are modus ponendo ponens (modus ponens) ax-mp 5, modus tollendo tollens (modus tollens) mto 634, modus ponendo tollens I mptnan 1384, modus ponendo tollens II mptxor 1385, and modus tollendo ponens (exclusive-or version) mtpxor 1387. The first is an axiom, the second is already proved; in this section we prove the other three. Since we assume or prove all of indemonstrables, the system of logic we use here is as at least as strong as the set of Stoic indemonstrables. Note that modus tollendo ponens mtpxor 1387 originally used exclusive-or, but over time the name modus tollendo ponens has increasingly referred to an inclusive-or variation, which is proved in mtpor 1386. This set of indemonstrables is not the entire system of Stoic logic.

 
Theoremmptnan 1384 Modus ponendo tollens 1, one of the "indemonstrables" in Stoic logic. See rule 1 on [Lopez-Astorga] p. 12 , rule 1 on [Sanford] p. 40, and rule A3 in [Hitchcock] p. 5. Sanford describes this rule second (after mptxor 1385) as a "safer, and these days much more common" version of modus ponendo tollens because it avoids confusion between inclusive-or and exclusive-or. (Contributed by David A. Wheeler, 3-Jul-2016.)
 |-  ph   &    |- 
 -.  ( ph  /\  ps )   =>    |- 
 -.  ps
 
Theoremmptxor 1385 Modus ponendo tollens 2, one of the "indemonstrables" in Stoic logic. Note that this uses exclusive-or  \/_. See rule 2 on [Lopez-Astorga] p. 12 , rule 4 on [Sanford] p. 39 and rule A4 in [Hitchcock] p. 5 . (Contributed by David A. Wheeler, 2-Mar-2018.)
 |-  ph   &    |-  ( ph  \/_  ps )   =>    |- 
 -.  ps
 
Theoremmtpor 1386 Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1387, one of the five original "indemonstrables" in Stoic logic. However, in Stoic logic this rule used exclusive-or, while the name modus tollendo ponens often refers to a variant of the rule that uses inclusive-or instead. The rule says, "if  ph is not true, and  ph or  ps (or both) are true, then  ps must be true." An alternate phrasing is, "Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth." -- Sherlock Holmes (Sir Arthur Conan Doyle, 1890: The Sign of the Four, ch. 6). (Contributed by David A. Wheeler, 3-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.)
 |- 
 -.  ph   &    |-  ( ph  \/  ps )   =>    |- 
 ps
 
Theoremmtpxor 1387 Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1386, one of the five "indemonstrables" in Stoic logic. The rule says, "if  ph is not true, and either  ph or  ps (exclusively) are true, then  ps must be true." Today the name "modus tollendo ponens" often refers to a variant, the inclusive-or version as defined in mtpor 1386. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1385, that is, it is exclusive-or df-xor 1337), rule 3 of [Sanford] p. 39 (where it is not as clearly stated which kind of "or" is used but it appears to be in the same sense as mptxor 1385), and rule A5 in [Hitchcock] p. 5 (exclusive-or is expressly used). (Contributed by David A. Wheeler, 4-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.) (Proof shortened by BJ, 19-Apr-2019.)
 |- 
 -.  ph   &    |-  ( ph  \/_  ps )   =>    |- 
 ps
 
Theoremstoic2a 1388 Stoic logic Thema 2 version a.

Statement T2 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic thema 2 as follows: "When from two assertibles a third follows, and from the third and one (or both) of the two another follows, then this other follows from the first two."

Bobzien uses constructs such as  ph, 
ps |-  ch; in Metamath we will represent that construct as  ph 
/\  ps  ->  ch.

This version a is without the phrase "or both"; see stoic2b 1389 for the version with the phrase "or both". We already have this rule as syldan 278, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)

 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ps )  ->  th )
 
Theoremstoic2b 1389 Stoic logic Thema 2 version b. See stoic2a 1388.

Version b is with the phrase "or both". We already have this rule as mpd3an3 1299, so here we prove the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)

 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ps )  ->  th )
 
Theoremstoic3 1390 Stoic logic Thema 3.

Statement T3 of [Bobzien] p. 116-117 discusses Stoic logic thema 3.

"When from two (assemblies) a third follows, and from the one that follows (i.e., the third) together with another, external external assumption, another follows, then other follows from the first two and the externally co-assumed one. (Simp. Cael. 237.2-4)" (Contributed by David A. Wheeler, 17-Feb-2019.)

 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ch 
 /\  th )  ->  ta )   =>    |-  (
 ( ph  /\  ps  /\  th )  ->  ta )
 
Theoremstoic4a 1391 Stoic logic Thema 4 version a.

Statement T4 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic thema 4: "When from two assertibles a third follows, and from the third and one (or both) of the two and one (or more) external assertible(s) another follows, then this other follows from the first two and the external(s)."

We use  th to represent the "external" assertibles. This is version a, which is without the phrase "or both"; see stoic4b 1392 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.)

 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ch 
 /\  ph  /\  th )  ->  ta )   =>    |-  ( ( ph  /\  ps  /\ 
 th )  ->  ta )
 
Theoremstoic4b 1392 Stoic logic Thema 4 version b.

This is version b, which is with the phrase "or both". See stoic4a 1391 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.)

 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ( ch  /\  ph  /\  ps )  /\  th )  ->  ta )   =>    |-  ( ( ph  /\  ps  /\ 
 th )  ->  ta )
 
1.2.16  Logical implication (continued)
 
Theoremsyl6an 1393 A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th ) )   &    |-  ( ( ps 
 /\  th )  ->  ta )   =>    |-  ( ph  ->  ( ch  ->  ta ) )
 
Theoremsyl10 1394 A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  ( th  ->  ta )
 ) )   &    |-  ( ch  ->  ( ta  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( th  ->  et ) ) )
 
Theoremexbir 1395 Exportation implication also converting head from biconditional to conditional. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  ( ( ( ph  /\ 
 ps )  ->  ( ch 
 <-> 
 th ) )  ->  ( ph  ->  ( ps  ->  ( th  ->  ch )
 ) ) )
 
Theorem3impexp 1396 impexp 261 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  ->  th )  <->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
 
Theorem3impexpbicom 1397 3impexp 1396 with biconditional consequent of antecedent that is commuted in consequent. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  ->  ( th  <->  ta ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) ) )
 
Theorem3impexpbicomi 1398 Deduction form of 3impexpbicom 1397. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ( th 
 <->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )
 
Theoremancomsimp 1399 Closed form of ancoms 266. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  ( ( ( ph  /\ 
 ps )  ->  ch )  <->  ( ( ps  /\  ph )  ->  ch ) )
 
Theoremexpcomd 1400 Deduction form of expcom 115. (Contributed by Alan Sare, 22-Jul-2012.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ph  ->  ( ch  ->  ( ps  ->  th ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13083
  Copyright terms: Public domain < Previous  Next >