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Type | Label | Description |
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Statement | ||
Theorem | syl332anc 1201 | Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) |
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Theorem | syl333anc 1202 | A syllogism inference combined with contraction. (Contributed by NM, 10-Mar-2012.) |
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Theorem | syl3an1 1203 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
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Theorem | syl3an2 1204 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
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Theorem | syl3an3 1205 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
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Theorem | syl3an1b 1206 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
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Theorem | syl3an2b 1207 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
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Theorem | syl3an3b 1208 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
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Theorem | syl3an1br 1209 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
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Theorem | syl3an2br 1210 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
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Theorem | syl3an3br 1211 | A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
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Theorem | syl3an 1212 | A triple syllogism inference. (Contributed by NM, 13-May-2004.) |
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Theorem | syl3anb 1213 | A triple syllogism inference. (Contributed by NM, 15-Oct-2005.) |
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Theorem | syl3anbr 1214 | A triple syllogism inference. (Contributed by NM, 29-Dec-2011.) |
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Theorem | syld3an3 1215 | A syllogism inference. (Contributed by NM, 20-May-2007.) |
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Theorem | syld3an1 1216 | A syllogism inference. (Contributed by NM, 7-Jul-2008.) |
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Theorem | syld3an2 1217 | A syllogism inference. (Contributed by NM, 20-May-2007.) |
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Theorem | syl3anl1 1218 | A syllogism inference. (Contributed by NM, 24-Feb-2005.) |
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Theorem | syl3anl2 1219 | A syllogism inference. (Contributed by NM, 24-Feb-2005.) |
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Theorem | syl3anl3 1220 | A syllogism inference. (Contributed by NM, 24-Feb-2005.) |
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Theorem | syl3anl 1221 | A triple syllogism inference. (Contributed by NM, 24-Dec-2006.) |
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Theorem | syl3anr1 1222 | A syllogism inference. (Contributed by NM, 31-Jul-2007.) |
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Theorem | syl3anr2 1223 | A syllogism inference. (Contributed by NM, 1-Aug-2007.) |
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Theorem | syl3anr3 1224 | A syllogism inference. (Contributed by NM, 23-Aug-2007.) |
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Theorem | 3impdi 1225 | Importation inference (undistribute conjunction). (Contributed by NM, 14-Aug-1995.) |
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Theorem | 3impdir 1226 | Importation inference (undistribute conjunction). (Contributed by NM, 20-Aug-1995.) |
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Theorem | 3anidm12 1227 | Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.) |
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Theorem | 3anidm13 1228 | Inference from idempotent law for conjunction. (Contributed by NM, 7-Mar-2008.) |
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Theorem | 3anidm23 1229 | Inference from idempotent law for conjunction. (Contributed by NM, 1-Feb-2007.) |
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Theorem | syl2an3an 1230 | syl3an 1212 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.) |
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Theorem | syl2an23an 1231 | Deduction related to syl3an 1212 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.) |
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Theorem | 3ori 1232 | Infer implication from triple disjunction. (Contributed by NM, 26-Sep-2006.) |
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Theorem | 3jao 1233 | Disjunction of 3 antecedents. (Contributed by NM, 8-Apr-1994.) |
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Theorem | 3jaob 1234 | Disjunction of 3 antecedents. (Contributed by NM, 13-Sep-2011.) |
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Theorem | 3jaoi 1235 | Disjunction of 3 antecedents (inference). (Contributed by NM, 12-Sep-1995.) |
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Theorem | 3jaod 1236 | Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.) |
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Theorem | 3jaoian 1237 | Disjunction of 3 antecedents (inference). (Contributed by NM, 14-Oct-2005.) |
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Theorem | 3jaodan 1238 | Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.) |
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Theorem | mpjao3dan 1239 | Eliminate a 3-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
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Theorem | 3jaao 1240 | Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
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Theorem | 3ianorr 1241 | Triple disjunction implies negated triple conjunction. (Contributed by Jim Kingdon, 23-Dec-2018.) |
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Theorem | syl3an9b 1242 | Nested syllogism inference conjoining 3 dissimilar antecedents. (Contributed by NM, 1-May-1995.) |
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Theorem | 3orbi123d 1243 | Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.) |
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Theorem | 3anbi123d 1244 | Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.) |
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Theorem | 3anbi12d 1245 | Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) |
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Theorem | 3anbi13d 1246 | Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) |
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Theorem | 3anbi23d 1247 | Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) |
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Theorem | 3anbi1d 1248 | Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
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Theorem | 3anbi2d 1249 | Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
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Theorem | 3anbi3d 1250 | Deduction adding conjuncts to an equivalence. (Contributed by NM, 8-Sep-2006.) |
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Theorem | 3anim123d 1251 | Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.) |
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Theorem | 3orim123d 1252 | Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.) |
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Theorem | an6 1253 | Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.) |
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Theorem | 3an6 1254 | Analog of an4 551 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
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Theorem | 3or6 1255 | Analog of or4 721 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) |
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Theorem | mp3an1 1256 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
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Theorem | mp3an2 1257 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
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Theorem | mp3an3 1258 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
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Theorem | mp3an12 1259 | An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.) |
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Theorem | mp3an13 1260 | An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) |
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Theorem | mp3an23 1261 | An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) |
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Theorem | mp3an1i 1262 | An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.) |
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Theorem | mp3anl1 1263 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
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Theorem | mp3anl2 1264 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
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Theorem | mp3anl3 1265 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
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Theorem | mp3anr1 1266 | An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.) |
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Theorem | mp3anr2 1267 | An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.) |
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Theorem | mp3anr3 1268 | An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.) |
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Theorem | mp3an 1269 | An inference based on modus ponens. (Contributed by NM, 14-May-1999.) |
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Theorem | mpd3an3 1270 | An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.) |
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Theorem | mpd3an23 1271 | An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.) |
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Theorem | mp3and 1272 | A deduction based on modus ponens. (Contributed by Mario Carneiro, 24-Dec-2016.) |
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Theorem | mp3an12i 1273 | mp3an 1269 with antecedents in standard conjunction form and with one hypothesis an implication. (Contributed by Alan Sare, 28-Aug-2016.) |
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Theorem | mp3an2i 1274 | mp3an 1269 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.) |
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Theorem | mp3an3an 1275 | mp3an 1269 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.) |
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Theorem | mp3an2ani 1276 | An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) |
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Theorem | biimp3a 1277 | Infer implication from a logical equivalence. Similar to biimpa 290. (Contributed by NM, 4-Sep-2005.) |
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Theorem | biimp3ar 1278 | Infer implication from a logical equivalence. Similar to biimpar 291. (Contributed by NM, 2-Jan-2009.) |
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Theorem | 3anandis 1279 | Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.) |
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Theorem | 3anandirs 1280 | Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.) (Revised by NM, 18-Apr-2007.) |
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Theorem | ecased 1281 | Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.) |
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Theorem | ecase23d 1282 | Variation of ecased 1281 with three disjuncts instead of two. (Contributed by NM, 22-Apr-1994.) (Revised by Jim Kingdon, 9-Dec-2017.) |
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Even though it isn't ordinarily part of propositional calculus, the universal
quantifier | ||
Syntax | wal 1283 |
Extend wff definition to include the universal quantifier ('for all').
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Even though it isn't ordinarily part of propositional calculus, the equality
predicate | ||
Syntax | cv 1284 |
This syntax construction states that a variable ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() While it is tempting and perhaps occasionally useful to view cv 1284 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1284 is intrinsically no different from any other class-building syntax such as cab 2069, cun 2980, or c0 3267. For a general discussion of the theory of classes and the role of cv 1284, see http://us.metamath.org/mpeuni/mmset.html#class.
(The description above applies to set theory, not predicate calculus.
The purpose of introducing |
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Syntax | wceq 1285 |
Extend wff definition to include class equality.
For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class.
(The purpose of introducing |
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Syntax | wtru 1286 |
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Theorem | trujust 1287 | Soundness justification theorem for df-tru 1288. (Contributed by Mario Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.) |
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Definition | df-tru 1288 |
Definition of the truth value "true", or "verum", denoted
by ![]() ![]() |
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Theorem | tru 1289 |
The truth value ![]() |
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Syntax | wfal 1290 |
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Definition | df-fal 1291 |
Definition of the truth value "false", or "falsum", denoted
by ![]() |
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Theorem | fal 1292 |
The truth value ![]() |
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Theorem | dftru2 1293 | An alternate definition of "true". (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) (New usage is discouraged.) |
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Theorem | trud 1294 |
Eliminate ![]() ![]() |
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Theorem | tbtru 1295 |
A proposition is equivalent to itself being equivalent to ![]() |
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Theorem | nbfal 1296 |
The negation of a proposition is equivalent to itself being equivalent to
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Theorem | bitru 1297 | A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.) |
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Theorem | bifal 1298 | A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.) |
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Theorem | falim 1299 |
The truth value ![]() |
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Theorem | falimd 1300 |
The truth value ![]() |
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