HomeHome Intuitionistic Logic Explorer
Theorem List (p. 14 of 152)
< 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
 
Theoremsyl3anr2 1301 A syllogism inference. (Contributed by NM, 1-Aug-2007.)
 |-  ( ph  ->  th )   &    |-  (
 ( ch  /\  ( ps  /\  th  /\  ta ) )  ->  et )   =>    |-  (
 ( ch  /\  ( ps  /\  ph  /\  ta )
 )  ->  et )
 
Theoremsyl3anr3 1302 A syllogism inference. (Contributed by NM, 23-Aug-2007.)
 |-  ( ph  ->  ta )   &    |-  (
 ( ch  /\  ( ps  /\  th  /\  ta ) )  ->  et )   =>    |-  (
 ( ch  /\  ( ps  /\  th  /\  ph )
 )  ->  et )
 
Theorem3impdi 1303 Importation inference (undistribute conjunction). (Contributed by NM, 14-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ph  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorem3impdir 1304 Importation inference (undistribute conjunction). (Contributed by NM, 20-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  ps ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ch  /\ 
 ps )  ->  th )
 
Theorem3anidm12 1305 Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
 |-  ( ( ph  /\  ph  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theorem3anidm13 1306 Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.)
 |-  ( ( ph  /\  ps  /\  ph )  ->  ch )   =>    |-  (
 ( ph  /\  ps )  ->  ch )
 
Theorem3anidm23 1307 Inference from idempotent law for conjunction. (Contributed by NM, 1-Feb-2007.)
 |-  ( ( ph  /\  ps  /\ 
 ps )  ->  ch )   =>    |-  (
 ( ph  /\  ps )  ->  ch )
 
Theoremsyl2an3an 1308 syl3an 1290 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( th  ->  ta )   &    |-  ( ( ps 
 /\  ch  /\  ta )  ->  et )   =>    |-  ( ( ph  /\  th )  ->  et )
 
Theoremsyl2an23an 1309 Deduction related to syl3an 1290 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ( th  /\  ph )  ->  ta )   &    |-  (
 ( ps  /\  ch  /\ 
 ta )  ->  et )   =>    |-  (
 ( th  /\  ph )  ->  et )
 
Theorem3ori 1310 Infer implication from triple disjunction. (Contributed by NM, 26-Sep-2006.)
 |-  ( ph  \/  ps  \/  ch )   =>    |-  ( ( -.  ph  /\ 
 -.  ps )  ->  ch )
 
Theorem3jao 1311 Disjunction of 3 antecedents. (Contributed by NM, 8-Apr-1994.)
 |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  ps )  /\  ( th  ->  ps ) )  ->  ( ( ph  \/  ch 
 \/  th )  ->  ps )
 )
 
Theorem3jaob 1312 Disjunction of 3 antecedents. (Contributed by NM, 13-Sep-2011.)
 |-  ( ( ( ph  \/  ch  \/  th )  ->  ps )  <->  ( ( ph  ->  ps )  /\  ( ch  ->  ps )  /\  ( th  ->  ps ) ) )
 
Theorem3jaoi 1313 Disjunction of 3 antecedents (inference). (Contributed by NM, 12-Sep-1995.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  ps )   &    |-  ( th  ->  ps )   =>    |-  ( ( ph  \/  ch 
 \/  th )  ->  ps )
 
Theorem3jaod 1314 Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  ->  ch ) )   &    |-  ( ph  ->  ( ta  ->  ch )
 )   =>    |-  ( ph  ->  (
 ( ps  \/  th  \/  ta )  ->  ch )
 )
 
Theorem3jaoian 1315 Disjunction of 3 antecedents (inference). (Contributed by NM, 14-Oct-2005.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( th  /\ 
 ps )  ->  ch )   &    |-  (
 ( ta  /\  ps )  ->  ch )   =>    |-  ( ( ( ph  \/  th  \/  ta )  /\  ps )  ->  ch )
 
Theorem3jaodan 1316 Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 th )  ->  ch )   &    |-  (
 ( ph  /\  ta )  ->  ch )   =>    |-  ( ( ph  /\  ( ps  \/  th  \/  ta ) )  ->  ch )
 
Theoremmpjao3dan 1317 Eliminate a 3-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 th )  ->  ch )   &    |-  (
 ( ph  /\  ta )  ->  ch )   &    |-  ( ph  ->  ( ps  \/  th  \/  ta ) )   =>    |-  ( ph  ->  ch )
 
Theorem3jaao 1318 Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ta  ->  ch ) )   &    |-  ( et  ->  ( ze  ->  ch )
 )   =>    |-  ( ( ph  /\  th  /\ 
 et )  ->  (
 ( ps  \/  ta  \/  ze )  ->  ch )
 )
 
Theorem3ianorr 1319 Triple disjunction implies negated triple conjunction. (Contributed by Jim Kingdon, 23-Dec-2018.)
 |-  ( ( -.  ph  \/  -.  ps  \/  -.  ch )  ->  -.  ( ph  /\  ps  /\  ch ) )
 
Theoremsyl3an9b 1320 Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ch  <->  ta ) )   &    |-  ( et  ->  ( ta  <->  ze ) )   =>    |-  ( ( ph  /\ 
 th  /\  et )  ->  ( ps  <->  ze ) )
 
Theorem3orbi123d 1321 Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   &    |-  ( ph  ->  ( et  <->  ze ) )   =>    |-  ( ph  ->  ( ( ps  \/  th  \/  et )  <->  ( ch  \/  ta 
 \/  ze ) ) )
 
Theorem3anbi123d 1322 Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   &    |-  ( ph  ->  ( et  <->  ze ) )   =>    |-  ( ph  ->  ( ( ps  /\  th  /\ 
 et )  <->  ( ch  /\  ta 
 /\  ze ) ) )
 
Theorem3anbi12d 1323 Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ( ps  /\  th  /\ 
 et )  <->  ( ch  /\  ta 
 /\  et ) ) )
 
Theorem3anbi13d 1324 Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ( ps  /\  et  /\ 
 th )  <->  ( ch  /\  et  /\  ta ) ) )
 
Theorem3anbi23d 1325 Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ( et  /\  ps  /\ 
 th )  <->  ( et  /\  ch 
 /\  ta ) ) )
 
Theorem3anbi1d 1326 Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( ps  /\  th  /\ 
 ta )  <->  ( ch  /\  th 
 /\  ta ) ) )
 
Theorem3anbi2d 1327 Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( th  /\  ps  /\ 
 ta )  <->  ( th  /\  ch 
 /\  ta ) ) )
 
Theorem3anbi3d 1328 Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( ( th  /\  ta  /\ 
 ps )  <->  ( th  /\  ta 
 /\  ch ) ) )
 
Theorem3anim123d 1329 Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  ->  ta ) )   &    |-  ( ph  ->  ( et  ->  ze )
 )   =>    |-  ( ph  ->  (
 ( ps  /\  th  /\ 
 et )  ->  ( ch  /\  ta  /\  ze ) ) )
 
Theorem3orim123d 1330 Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  ->  ta ) )   &    |-  ( ph  ->  ( et  ->  ze )
 )   =>    |-  ( ph  ->  (
 ( ps  \/  th  \/  et )  ->  ( ch  \/  ta  \/  ze ) ) )
 
Theoreman6 1331 Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.)
 |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  ( th  /\  ta  /\ 
 et ) )  <->  ( ( ph  /\ 
 th )  /\  ( ps  /\  ta )  /\  ( ch  /\  et )
 ) )
 
Theorem3an6 1332 Analog of an4 586 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th )  /\  ( ta  /\  et )
 ) 
 <->  ( ( ph  /\  ch  /\ 
 ta )  /\  ( ps  /\  th  /\  et ) ) )
 
Theorem3or6 1333 Analog of or4 772 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.)
 |-  ( ( ( ph  \/  ps )  \/  ( ch  \/  th )  \/  ( ta  \/  et ) )  <->  ( ( ph  \/  ch  \/  ta )  \/  ( ps  \/  th  \/  et ) ) )
 
Theoremmp3an1 1334 An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
 |-  ph   &    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ps  /\  ch )  ->  th )
 
Theoremmp3an2 1335 An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
 |- 
 ps   &    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ch )  ->  th )
 
Theoremmp3an3 1336 An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.)
 |- 
 ch   &    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  (
 ( ph  /\  ps )  ->  th )
 
Theoremmp3an12 1337 An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.)
 |-  ph   &    |- 
 ps   &    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  ( ch  ->  th )
 
Theoremmp3an13 1338 An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
 |-  ph   &    |- 
 ch   &    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  ( ps  ->  th )
 
Theoremmp3an23 1339 An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.)
 |- 
 ps   &    |- 
 ch   &    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremmp3an1i 1340 An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.)
 |- 
 ps   &    |-  ( ph  ->  (
 ( ps  /\  ch  /\ 
 th )  ->  ta )
 )   =>    |-  ( ph  ->  (
 ( ch  /\  th )  ->  ta ) )
 
Theoremmp3anl1 1341 An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
 |-  ph   &    |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ta )   =>    |-  (
 ( ( ps  /\  ch )  /\  th )  ->  ta )
 
Theoremmp3anl2 1342 An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
 |- 
 ps   &    |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ta )   =>    |-  (
 ( ( ph  /\  ch )  /\  th )  ->  ta )
 
Theoremmp3anl3 1343 An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.)
 |- 
 ch   &    |-  ( ( ( ph  /\ 
 ps  /\  ch )  /\  th )  ->  ta )   =>    |-  (
 ( ( ph  /\  ps )  /\  th )  ->  ta )
 
Theoremmp3anr1 1344 An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.)
 |- 
 ps   &    |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  ta )   =>    |-  (
 ( ph  /\  ( ch 
 /\  th ) )  ->  ta )
 
Theoremmp3anr2 1345 An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.)
 |- 
 ch   &    |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  ta )   =>    |-  (
 ( ph  /\  ( ps 
 /\  th ) )  ->  ta )
 
Theoremmp3anr3 1346 An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.)
 |- 
 th   &    |-  ( ( ph  /\  ( ps  /\  ch  /\  th ) )  ->  ta )   =>    |-  (
 ( ph  /\  ( ps 
 /\  ch ) )  ->  ta )
 
Theoremmp3an 1347 An inference based on modus ponens. (Contributed by NM, 14-May-1999.)
 |-  ph   &    |- 
 ps   &    |- 
 ch   &    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )   =>    |-  th
 
Theoremmpd3an3 1348 An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )   =>    |-  ( ( ph  /\  ps )  ->  th )
 
Theoremmpd3an23 1349 An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )   =>    |-  ( ph  ->  th )
 
Theoremmp3and 1350 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 1351 mp3an 1347 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 1352 mp3an 1347 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 1353 mp3an 1347 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 1354 An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  ph   &    |-  ( ps  ->  ch )   &    |-  (
 ( ps  /\  th )  ->  ta )   &    |-  ( ( ph  /\ 
 ch  /\  ta )  ->  et )   =>    |-  ( ( ps  /\  th )  ->  et )
 
Theorembiimp3a 1355 Infer implication from a logical equivalence. Similar to biimpa 296. (Contributed by NM, 4-Sep-2005.)
 |-  ( ( ph  /\  ps )  ->  ( ch  <->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembiimp3ar 1356 Infer implication from a logical equivalence. Similar to biimpar 297. (Contributed by NM, 2-Jan-2009.)
 |-  ( ( ph  /\  ps )  ->  ( ch  <->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  th )  ->  ch )
 
Theorem3anandis 1357 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 1358 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 1359 Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.)
 |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  ( ps  \/  ch ) )   =>    |-  ( ph  ->  ps )
 
Theoremecase23d 1360 Variation of ecased 1359 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 is not ordinarily part of propositional calculus, the universal quantifier  A. is introduced here so that the soundness of Definition df-tru 1366 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in Axiom ax-5 1457 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 1371 may be adopted and this subsection moved down to the start of the subsection with wex 1502 below. However, the use of dftru2 1371 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

 
Syntaxwal 1361 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 is not ordinarily part of propositional calculus, the equality predicate  = is introduced here so that the soundness of definition df-tru 1366 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in Axiom ax-8 1514 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 1371 may be adopted and this subsection moved down to just above weq 1513 below. However, the use of dftru2 1371 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid.

 
Syntaxcv 1362 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 2173. Since (when  y is distinct from  x) we have  x  =  { y  |  y  e.  x } by cvjust 2182, 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 1362 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1362 is intrinsically no different from any other class-building syntax such as cab 2173, cun 3139, or c0 3434.

For a general discussion of the theory of classes and the role of cv 1362, see https://us.metamath.org/mpeuni/mmset.html#class 1362.

(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 1513 of predicate calculus from the wceq 1363 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 1363 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 1513 of predicate calculus in terms of the wceq 1363 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 1513 or wceq 1363, 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 2180 for more information on the set theory usage of wceq 1363.)

 wff  A  =  B
 
1.2.12.3  Define the true and false constants
 
Syntaxwtru 1364 T. is a wff.
 wff T.
 
Theoremtrujust 1365 Soundness justification theorem for df-tru 1366. (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 1366 Definition of the truth value "true", or "verum", denoted by T.. This is a tautology, as proved by tru 1367. 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 1367, and other proofs should depend on tru 1367 (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 1367 The truth value T. is provable. (Contributed by Anthony Hart, 13-Oct-2010.)
 |- T.
 
Syntaxwfal 1368 F. is a wff.
 wff F.
 
Definitiondf-fal 1369 Definition of the truth value "false", or "falsum", denoted by F.. See also df-tru 1366. (Contributed by Anthony Hart, 22-Oct-2010.)
 |-  ( F.  <->  -. T.  )
 
Theoremfal 1370 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 1371 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 1372 Eliminate T. as an antecedent. A proposition implied by T. is true. (Contributed by Mario Carneiro, 13-Mar-2014.)
 |-  ( T.  ->  ph )   =>    |-  ph
 
Theoremtbtru 1373 A proposition is equivalent to itself being equivalent to T.. (Contributed by Anthony Hart, 14-Aug-2011.)
 |-  ( ph  <->  ( ph  <-> T.  ) )
 
Theoremnbfal 1374 The negation of a proposition is equivalent to itself being equivalent to F.. (Contributed by Anthony Hart, 14-Aug-2011.)
 |-  ( -.  ph  <->  ( ph  <-> F.  ) )
 
Theorembitru 1375 A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.)
 |-  ph   =>    |-  ( ph  <-> T.  )
 
Theorembifal 1376 A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.)
 |- 
 -.  ph   =>    |-  ( ph  <-> F.  )
 
Theoremfalim 1377 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 1378 The truth value F. implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ( ph  /\ F.  )  ->  ps )
 
Theoremtrud 1379 Anything implies T.. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.)
 |-  ( ph  -> T.  )
 
Theoremtruan 1380 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 1381 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 1382 Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ( ph  /\  ps )  -> F.  )   =>    |-  ( ph  ->  -. 
 ps )
 
Theorempm2.21fal 1383 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 1384 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 1385 Extend wff definition to include exclusive disjunction ('xor').
 wff  ( ph  \/_  ps )
 
Definitiondf-xor 1386 Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. Contrast with  /\ (wa 104),  \/ (wo 709), and  -> (wi 4) . (Contributed by FL, 22-Nov-2010.) (Modified by Jim Kingdon, 1-Mar-2018.)
 |-  ( ( ph  \/_  ps ) 
 <->  ( ( ph  \/  ps )  /\  -.  ( ph  /\  ps ) ) )
 
Theoremxoranor 1387 One way of defining exclusive or. Equivalent to df-xor 1386. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.)
 |-  ( ( ph  \/_  ps ) 
 <->  ( ( ph  \/  ps )  /\  ( -.  ph  \/  -.  ps )
 ) )
 
Theoremexcxor 1388 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 1389 XOR implies OR. (Contributed by BJ, 19-Apr-2019.)
 |-  ( ( ph  \/_  ps )  ->  ( ph  \/  ps ) )
 
Theoremxorbi2d 1390 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 1391 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 1392 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 1393 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  \/_  ch ) 
 <->  ( ps  \/_  th )
 )
 
Theoremxorbin 1394 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 1395 One direction of pm5.18dc 884, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.)
 |-  ( ( ph  <->  ps )  ->  -.  ( ph 
 <->  -.  ps ) )
 
Theoremxornbi 1396 A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1401. (Contributed by Jim Kingdon, 10-Mar-2018.)
 |-  ( ( ph  \/_  ps )  ->  -.  ( ph  <->  ps ) )
 
Theoremxor3dc 1397 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 1398  \/_ is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.)
 |-  ( ( ph  \/_  ps ) 
 <->  ( ps  \/_  ph )
 )
 
Theorempm5.15dc 1399 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 1400 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 )
 ) ) ) )
    < 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 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15104
  Copyright terms: Public domain < Previous  Next >