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Theorem List for Metamath Proof Explorer - 2301-2400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremeqtr2 2301 A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremeqtr3 2302 A transitive law for class equality. (Contributed by NM, 20-May-2005.)

Theoremeqtri 2303 An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)

Theoremeqtr2i 2304 An equality transitivity inference. (Contributed by NM, 21-Feb-1995.)

Theoremeqtr3i 2305 An equality transitivity inference. (Contributed by NM, 6-May-1994.)

Theoremeqtr4i 2306 An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)

Theorem3eqtri 2307 An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.)

Theorem3eqtrri 2308 An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem3eqtr2i 2309 An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.)

Theorem3eqtr2ri 2310 An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem3eqtr3i 2311 An inference from three chained equalities. (Contributed by NM, 6-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem3eqtr3ri 2312 An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.)

Theorem3eqtr4i 2313 An inference from three chained equalities. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem3eqtr4ri 2314 An inference from three chained equalities. (Contributed by NM, 2-Sep-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremeqtrd 2315 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)

Theoremeqtr2d 2316 An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.)

Theoremeqtr3d 2317 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)

Theoremeqtr4d 2318 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)

Theorem3eqtrd 2319 A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.)

Theorem3eqtrrd 2320 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem3eqtr2d 2321 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)

Theorem3eqtr2rd 2322 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)

Theorem3eqtr3d 2323 A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem3eqtr3rd 2324 A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.)

Theorem3eqtr4d 2325 A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem3eqtr4rd 2326 A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995.)

Theoremsyl5eq 2327 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)

Theoremsyl5req 2328 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)

Theoremsyl5eqr 2329 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)

Theoremsyl5reqr 2330 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)

Theoremsyl6eq 2331 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)

Theoremsyl6req 2332 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)

Theoremsyl6eqr 2333 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)

Theoremsyl6reqr 2334 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)

Theoremsylan9eq 2335 An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremsylan9req 2336 An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)

Theoremsylan9eqr 2337 An equality transitivity deduction. (Contributed by NM, 8-May-1994.)

Theorem3eqtr3g 2338 A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)

Theorem3eqtr3a 2339 A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.)

Theorem3eqtr4g 2340 A chained equality inference, useful for converting to definitions. (Contributed by NM, 5-Aug-1993.)

Theorem3eqtr4a 2341 A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremeq2tri 2342 A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)

Theoremeleq1 2343 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq2 2344 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq12 2345 Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)

Theoremeleq1i 2346 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq2i 2347 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq12i 2348 Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)

Theoremeleq1d 2349 Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq2d 2350 Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.)

Theoremeleq12d 2351 Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)

Theoremeleq1a 2352 A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)

Theoremeqeltri 2353 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Theoremeqeltrri 2354 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Theoremeleqtri 2355 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Theoremeleqtrri 2356 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Theoremeqeltrd 2357 Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremeqeltrrd 2358 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)

Theoremeleqtrd 2359 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)

Theoremeleqtrrd 2360 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)

Theorem3eltr3i 2361 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr4i 2362 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr3d 2363 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr4d 2364 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr3g 2365 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr4g 2366 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theoremsyl5eqel 2367 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl5eqelr 2368 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl5eleq 2369 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl5eleqr 2370 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eqel 2371 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eqelr 2372 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eleq 2373 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eleqr 2374 A membership and equality inference. (Contributed by NM, 24-Apr-2005.)

Theoremeleq2s 2375 Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremeqneltrd 2376 If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremeqneltrrd 2377 If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremneleqtrd 2378 If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremneleqtrrd 2379 If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremcleqh 2380* Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 5-Aug-1993.)

Theoremnelneq 2381 A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)

Theoremnelneq2 2382 A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)

Theoremeqsb3lem 2383* Lemma for eqsb3 2384. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremeqsb3 2384* Substitution applied to an atomic wff (class version of equsb3 2041). (Contributed by Rodolfo Medina, 28-Apr-2010.)

Theoremclelsb3 2385* Substitution applied to an atomic wff (class version of elsb3 2042). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremhbxfreq 2386 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1555 for equivalence version. (Contributed by NM, 21-Aug-2007.)

Theoremhblem 2387* Change the free variable of a hypothesis builder. Lemma for nfcrii 2412. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)

Theoremabeq2 2388* Equality of a class variable and a class abstraction (also called a class builder). Theorem 5.1 of [Quine] p. 34. This theorem shows the relationship between expressions with class abstractions and expressions with class variables. Note that abbi 2393 and its relatives are among those useful for converting theorems with class variables to equivalent theorems with wff variables, by first substituting a class abstraction for each class variable.

Class variables can always be eliminated from a theorem to result in an equivalent theorem with wff variables, and vice-versa. The idea is roughly as follows. To convert a theorem with a wff variable (that has a free variable ) to a theorem with a class variable , we substitute for throughout and simplify, where is a new class variable not already in the wff. An example is the conversion of zfauscl 4143 to inex1 4155 (look at the instance of zfauscl 4143 that occurs in the proof of inex1 4155). Conversely, to convert a theorem with a class variable to one with , we substitute for throughout and simplify, where and are new set and wff variables not already in the wff. An example is cp 7561, which derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 7560. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13. (Contributed by NM, 5-Aug-1993.)

Theoremabeq1 2389* Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)

Theoremabeq2i 2390 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 3-Apr-1996.)

Theoremabeq1i 2391 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.)

Theoremabeq2d 2392 Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)

Theoremabbi 2393 Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.)

Theoremabbi2i 2394* Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 5-Aug-1993.)

Theoremabbii 2395 Equivalent wff's yield equal class abstractions (inference rule). (Contributed by NM, 5-Aug-1993.)

Theoremabbid 2396 Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremabbidv 2397* Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 10-Aug-1993.)

Theoremabbi2dv 2398* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)

Theoremabbi1dv 2399* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)

Theoremabid2 2400* A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)

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