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

Theoremsyl6eq 2301 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)

Theoremsyl6req 2302 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)

Theoremsyl6eqr 2303 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)

Theoremsyl6reqr 2304 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)

Theoremsylan9eq 2305 An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremsylan9req 2306 An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)

Theoremsylan9eqr 2307 An equality transitivity deduction. (Contributed by NM, 8-May-1994.)

Theorem3eqtr3g 2308 A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)

Theorem3eqtr3a 2309 A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.)

Theorem3eqtr4g 2310 A chained equality inference, useful for converting to definitions. (Contributed by NM, 5-Aug-1993.)

Theorem3eqtr4a 2311 A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremeq2tri 2312 A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)

Theoremeleq1 2313 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq2 2314 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq12 2315 Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)

Theoremeleq1i 2316 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq2i 2317 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq12i 2318 Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)

Theoremeleq1d 2319 Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq2d 2320 Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.)

Theoremeleq12d 2321 Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)

Theoremeleq1a 2322 A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)

Theoremeqeltri 2323 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Theoremeqeltrri 2324 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Theoremeleqtri 2325 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Theoremeleqtrri 2326 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Theoremeqeltrd 2327 Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremeqeltrrd 2328 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)

Theoremeleqtrd 2329 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)

Theoremeleqtrrd 2330 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)

Theorem3eltr3i 2331 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr4i 2332 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr3d 2333 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr4d 2334 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr3g 2335 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr4g 2336 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theoremsyl5eqel 2337 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl5eqelr 2338 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl5eleq 2339 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl5eleqr 2340 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eqel 2341 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eqelr 2342 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eleq 2343 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eleqr 2344 A membership and equality inference. (Contributed by NM, 24-Apr-2005.)

Theoremeleq2s 2345 Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremcleqh 2346* Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 5-Aug-1993.)

Theoremnelneq 2347 A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)

Theoremnelneq2 2348 A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)

Theoremeqsb3lem 2349* Lemma for eqsb3 2350. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremeqsb3 2350* Substitution applied to an atomic wff (class version of equsb3 2062). (Contributed by Rodolfo Medina, 28-Apr-2010.)

Theoremclelsb3 2351* Substitution applied to an atomic wff (class version of elsb3 2063). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremhbxfreq 2352 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1560 for equivalence version. (Contributed by NM, 21-Aug-2007.)

Theoremhblem 2353* Change the free variable of a hypothesis builder. Lemma for nfcrii 2378. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)

Theoremabeq2 2354* 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 2359 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 4040 to inex1 4052 (look at the instance of zfauscl 4040 that occurs in the proof of inex1 4052). 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 7445, which derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 7444. 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 2355* Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)

Theoremabeq2i 2356 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 3-Apr-1996.)

Theoremabeq1i 2357 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.)

Theoremabeq2d 2358 Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)

Theoremabbi 2359 Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.)

Theoremabbi2i 2360* Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 5-Aug-1993.)

Theoremabbii 2361 Equivalent wff's yield equal class abstractions (inference rule). (Contributed by NM, 5-Aug-1993.)

Theoremabbid 2362 Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremabbidv 2363* Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 10-Aug-1993.)

Theoremabbi2dv 2364* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)

Theoremabbi1dv 2365* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)

Theoremabid2 2366* A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)

Theoremcbvab 2367 Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)

Theoremcbvabv 2368* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)

Theoremclelab 2369* Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)

Theoremclabel 2370* Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)

Theoremsbab 2371* The right-hand side of the second equality is a way of representing proper substitution of for into a class variable. (Contributed by NM, 14-Sep-2003.)

2.1.3  Class form not-free predicate

Syntaxwnfc 2372 Extend wff definition to include the not-free predicate for classes.

Theoremnfcjust 2373* Justification theorem for df-nfc 2374. (Contributed by Mario Carneiro, 13-Oct-2016.)

Definitiondf-nfc 2374* Define the not-free predicate for classes. This is read " is not free in ". Not-free means that the value of cannot affect the value of , e.g., any occurrence of in is effectively bound by a "for all" or something that expands to one (such as "there exists"). It is defined in terms of the not-free predicate df-nf 1540 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfci 2375* Deduce that a class does not have free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcii 2376* Deduce that a class does not have free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcr 2377* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcrii 2378* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcri 2379* Consequence of the not-free predicate. (Note that unlike nfcr 2377, this does not require and to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcd 2380* Deduce that a class does not have free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfceqi 2381 Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcxfr 2382 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcxfrd 2383 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfceqdf 2384 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremnfcv 2385* If is disjoint from , then is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfcvd 2386* If is disjoint from , then is not free in . (Contributed by Mario Carneiro, 7-Oct-2016.)

Theoremnfab1 2387 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfnfc1 2388 is bound in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfab 2389 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfaba1 2390 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremnfnfc 2391 Hypothesis builder for . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfeq 2392 Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfel 2393 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfeq1 2394* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)

Theoremnfel1 2395* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)

Theoremnfeq2 2396* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)

Theoremnfel2 2397* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)

Theoremnfcrd 2398* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremnfeqd 2399 Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)

Theoremnfeld 2400 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)

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