HomeHome Intuitionistic Logic Explorer
Theorem List (p. 14 of 114)
< 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
 
Theoremfalim 1301 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 1302 The truth value F. implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ( ph  /\ F.  )  ->  ps )
 
Theorema1tru 1303 Anything implies T.. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
 |-  ( ph  -> T.  )
 
Theoremtruan 1304 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 1305 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 1306 Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ( ph  /\  ps )  -> F.  )   =>    |-  ( ph  ->  -. 
 ps )
 
Theorempm2.21fal 1307 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 1308 Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof rewritten by Jim Kingdon, 4-May-2018.)
 |-  ( ( ph  <->  ( ps  /\  -.  ph ) )  ->  -.  ps )
 
1.2.14  Logical 'xor'
 
Syntaxwxo 1309 Extend wff definition to include exclusive disjunction ('xor').
 wff  ( ph  \/_  ps )
 
Definitiondf-xor 1310 Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. Contrast with  /\ (wa 102),  \/ (wo 662), and  -> (wi 4) . (Contributed by FL, 22-Nov-2010.) (Modified by Jim Kingdon, 1-Mar-2018.)
 |-  ( ( ph  \/_  ps ) 
 <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\  ps ) ) )
 
Theoremxoranor 1311 One way of defining exclusive or. Equivalent to df-xor 1310. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.)
 |-  ( ( ph  \/_  ps ) 
 <->  ( ( ph  \/  ps )  /\  ( -.  ph  \/  -.  ps )
 ) )
 
Theoremexcxor 1312 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 1313 XOR implies OR. (Contributed by BJ, 19-Apr-2019.)
 |-  ( ( ph  \/_  ps )  ->  ( ph  \/  ps ) )
 
Theoremxorbi2d 1314 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 1315 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 1316 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 1317 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  \/_  ch ) 
 <->  ( ps  \/_  th )
 )
 
Theoremxorbin 1318 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 1319 One direction of pm5.18dc 813, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.)
 |-  ( ( ph  <->  ps )  ->  -.  ( ph 
 <->  -.  ps ) )
 
Theoremxornbi 1320 A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1325. (Contributed by Jim Kingdon, 10-Mar-2018.)
 |-  ( ( ph  \/_  ps )  ->  -.  ( ph  <->  ps ) )
 
Theoremxor3dc 1321 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 1322  \/_ is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.)
 |-  ( ( ph  \/_  ps ) 
 <->  ( ps  \/_  ph )
 )
 
Theorempm5.15dc 1323 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 1324 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 1325 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 1326 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 1327 Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.)
 |-  ( ( ph  \/_  ps )  -> DECID  ph )
 
Theoremnbbndc 1328 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 1329 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 1330 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 1331 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 1332 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 1333 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 1334 Conjunction distributes over exclusive-or. (Contributed by Mario Carneiro and Jim Kingdon, 7-Oct-2018.)
 |-  ( ( ph  /\  ( ps  \/_  ch ) )  <-> 
 ( ( ph  /\  ps )  \/_  ( ph  /\  ch ) ) )
 
1.2.15  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 102,  \/ (disjunction aka logical inclusive 'or') wo 662,  -> (implies) wi 4,  -. (not) wn 3,  <-> (logical equivalence) df-bi 115, and  \/_ (exclusive or) df-xor 1310.

 
Theoremtruantru 1335 A  /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( ( T.  /\ T.  )  <-> T.  )
 
Theoremtruanfal 1336 A  /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( ( T.  /\ F.  )  <-> F.  )
 
Theoremfalantru 1337 A  /\ identity. (Contributed by David A. Wheeler, 23-Feb-2018.)
 |-  ( ( F.  /\ T.  )  <-> F.  )
 
Theoremfalanfal 1338 A  /\ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( ( F.  /\ F.  )  <-> F.  )
 
Theoremtruortru 1339 A  \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( T.  \/ T.  )  <-> T.  )
 
Theoremtruorfal 1340 A  \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( ( T.  \/ F.  )  <-> T.  )
 
Theoremfalortru 1341 A  \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( ( F.  \/ T.  )  <-> T.  )
 
Theoremfalorfal 1342 A  \/ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( F.  \/ F.  )  <-> F.  )
 
Theoremtruimtru 1343 A  -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( ( T.  -> T.  )  <-> T.  )
 
Theoremtruimfal 1344 A  -> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( T.  -> F.  )  <-> F.  )
 
Theoremfalimtru 1345 A  -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( ( F.  -> T.  )  <-> T.  )
 
Theoremfalimfal 1346 A  -> identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( ( F.  -> F.  )  <-> T.  )
 
Theoremnottru 1347 A  -. identity. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( -. T.  <-> F.  )
 
Theoremnotfal 1348 A  -. identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( -. F.  <-> T.  )
 
Theoremtrubitru 1349 A  <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( T.  <-> T.  )  <-> T.  )
 
Theoremtrubifal 1350 A  <-> identity. (Contributed by David A. Wheeler, 23-Feb-2018.)
 |-  ( ( T.  <-> F.  )  <-> F.  )
 
Theoremfalbitru 1351 A  <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( F.  <-> T.  )  <-> F.  )
 
Theoremfalbifal 1352 A  <-> identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ( F.  <-> F.  )  <-> T.  )
 
Theoremtruxortru 1353 A  \/_ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
 |-  ( ( T.  \/_ T.  )  <-> F.  )
 
Theoremtruxorfal 1354 A  \/_ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
 |-  ( ( T.  \/_ F.  )  <-> T.  )
 
Theoremfalxortru 1355 A  \/_ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
 |-  ( ( F.  \/_ T.  )  <-> T.  )
 
Theoremfalxorfal 1356 A  \/_ identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
 |-  ( ( F.  \/_ F.  )  <-> F.  )
 
1.2.16  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 7, modus tollendo tollens (modus tollens) mto 621, modus ponendo tollens I mptnan 1357, modus ponendo tollens II mptxor 1358, and modus tollendo ponens (exclusive-or version) mtpxor 1360. 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 1360 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 1359. This set of indemonstrables is not the entire system of Stoic logic.

 
Theoremmptnan 1357 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 1358) 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 1358 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 1359 Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1360, 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 1360 Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1359, 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 1359. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1358, that is, it is exclusive-or df-xor 1310), 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 1358), 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 1361 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 1362 for the version with the phrase "or both". We already have this rule as syldan 276, 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 1362 Stoic logic Thema 2 version b. See stoic2a 1361.

Version b is with the phrase "or both". We already have this rule as mpd3an3 1272, 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 1363 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 1364 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 1365 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 1365 Stoic logic Thema 4 version b.

This is version b, which is with the phrase "or both". See stoic4a 1364 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.)

 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ( ch  /\  ph  /\  ps )  /\  th )  ->  ta )   =>    |-  ( ( ph  /\  ps  /\ 
 th )  ->  ta )
 
1.2.17  Logical implication (continued)
 
Theoremsyl6an 1366 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 1367 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 1368 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 1369 impexp 259 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  ->  th )  <->  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) ) )
 
Theorem3impexpbicom 1370 3impexp 1369 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 1371 Deduction form of 3impexpbicom 1370. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  ( ( ph  /\  ps  /\ 
 ch )  ->  ( th 
 <->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )
 
Theoremancomsimp 1372 Closed form of ancoms 264. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  ( ( ( ph  /\ 
 ps )  ->  ch )  <->  ( ( ps  /\  ph )  ->  ch ) )
 
Theoremexpcomd 1373 Deduction form of expcom 114. (Contributed by Alan Sare, 22-Jul-2012.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ph  ->  ( ch  ->  ( ps  ->  th ) ) )
 
Theoremexpdcom 1374 Commuted form of expd 254. (Contributed by Alan Sare, 18-Mar-2012.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   =>    |-  ( ps  ->  ( ch  ->  ( ph  ->  th ) ) )
 
Theoremsimplbi2comg 1375 Implication form of simplbi2com 1376. (Contributed by Alan Sare, 22-Jul-2012.)
 |-  ( ( ph  <->  ( ps  /\  ch ) )  ->  ( ch  ->  ( ps  ->  ph ) ) )
 
Theoremsimplbi2com 1376 A deduction eliminating a conjunct, similar to simplbi2 377. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.)
 |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ch  ->  ( ps  ->  ph ) )
 
Theoremsyl6ci 1377 A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  th )   &    |-  ( ch  ->  ( th  ->  ta )
 )   =>    |-  ( ph  ->  ( ps  ->  ta ) )
 
Theoremmpisyl 1378 A syllogism combined with a modus ponens inference. (Contributed by Alan Sare, 25-Jul-2011.)
 |-  ( ph  ->  ps )   &    |-  ch   &    |-  ( ps  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  th )
 
1.3  Predicate calculus mostly without distinct variables
 
1.3.1  Universal quantifier (continued)

The universal quantifier was introduced above in wal 1285 for use by df-tru 1290. See the comments in that section. In this section, we continue with the first "real" use of it.

 
Axiomax-5 1379 Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
 
Axiomax-7 1380 Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. One of the predicate logic axioms which do not involve equality. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Axiomax-gen 1381 Rule of Generalization. The postulated inference rule of predicate calculus. See e.g. Rule 2 of [Hamilton] p. 74. This rule says that if something is unconditionally true, then it is true for all values of a variable. For example, if we have proved  x  =  x, we can conclude  A. x x  =  x or even  A. y
x  =  x. Theorem spi 1472 shows we can go the other way also: in other words we can add or remove universal quantifiers from the beginning of any theorem as required. (Contributed by NM, 5-Aug-1993.)
 |-  ph   =>    |- 
 A. x ph
 
Theoremgen2 1382 Generalization applied twice. (Contributed by NM, 30-Apr-1998.)
 |-  ph   =>    |- 
 A. x A. y ph
 
Theoremmpg 1383 Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.)
 |-  ( A. x ph  ->  ps )   &    |-  ph   =>    |- 
 ps
 
Theoremmpgbi 1384 Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
 |-  ( A. x ph  <->  ps )   &    |-  ph   =>    |- 
 ps
 
Theoremmpgbir 1385 Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
 |-  ( ph  <->  A. x ps )   &    |-  ps   =>    |-  ph
 
Theorema7s 1386 Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x A. y ph  ->  ps )   =>    |-  ( A. y A. x ph  ->  ps )
 
Theoremalimi 1387 Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x ph  ->  A. x ps )
 
Theorem2alimi 1388 Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x A. y ph  ->  A. x A. y ps )
 
Theoremalim 1389 Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 30-Mar-2008.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
 
Theoremal2imi 1390 Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( A. x ph 
 ->  ( A. x ps  ->  A. x ch )
 )
 
Theoremalanimi 1391 Variant of al2imi 1390 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( A. x ph 
 /\  A. x ps )  ->  A. x ch )
 
Syntaxwnf 1392 Extend wff definition to include the not-free predicate.
 wff  F/ x ph
 
Definitiondf-nf 1393 Define the not-free predicate for wffs. This is read " x is not free in  ph". Not-free means that the value of  x cannot affect the value of  ph, e.g., any occurrence of  x in  ph is effectively bound by a "for all" or something that expands to one (such as "there exists"). In particular, substitution for a variable not free in a wff does not affect its value (sbf 1704). An example of where this is used is stdpc5 1519. See nf2 1601 for an alternate definition which does not involve nested quantifiers on the same variable.

Not-free is a commonly used constraint, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the not-free notion within the logic itself rather than as a metalogical side condition.

To be precise, our definition really means "effectively not free," because it is slightly less restrictive than the usual textbook definition for not-free (which only considers syntactic freedom). For example,  x is effectively not free in the bare expression  x  =  x, even though  x would be considered free in the usual textbook definition, because the value of  x in the expression  x  =  x cannot affect the truth of the expression (and thus substitution will not change the result). (Contributed by Mario Carneiro, 11-Aug-2016.)

 |-  ( F/ x ph  <->  A. x ( ph  ->  A. x ph ) )
 
Theoremnfi 1394 Deduce that  x is not free in  ph from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  ->  A. x ph )   =>    |- 
 F/ x ph
 
Theoremhbth 1395 No variable is (effectively) free in a theorem.

This and later "hypothesis-building" lemmas, with labels starting "hb...", allow us to construct proofs of formulas of the form  |-  ( ph  ->  A. x ph ) from smaller formulas of this form. These are useful for constructing hypotheses that state " x is (effectively) not free in  ph." (Contributed by NM, 5-Aug-1993.)

 |-  ph   =>    |-  ( ph  ->  A. x ph )
 
Theoremnfth 1396 No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ph   =>    |- 
 F/ x ph
 
Theoremnfnth 1397 No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.)
 |- 
 -.  ph   =>    |- 
 F/ x ph
 
Theoremnftru 1398 The true constant has no free variables. (This can also be proven in one step with nfv 1464, but this proof does not use ax-17 1462.) (Contributed by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ x T.
 
Theoremalimdh 1399 Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 4-Jan-2002.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( A. x ps  ->  A. x ch ) )
 
Theoremalbi 1400 Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x (
 ph 
 <->  ps )  ->  ( A. x ph  <->  A. x ps )
 )
    < 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 11301-11394
  Copyright terms: Public domain < Previous  Next >