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Theorem List for Metamath Proof Explorer - 2401-2500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremeqid 2401 Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41.

This law is thought to have originated with Aristotle (Metaphysics, Zeta, 17, 1041 a, 10-20: "Therefore, inquiring why a thing is itself, it's inquiring nothing; ... saying that the thing is itself constitutes the sole reasoning and the sole cause, in every case, to the question of why the man is man or the musician musician."). (Thanks to Stefan Allan and Benoît Jubin for this information.) (Contributed by NM, 5-Aug-1993.) (Revised by Benoît Jubin, 14-Oct-2017.)

Theoremeqidd 2402 Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.)

Theoremeqcom 2403 Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 5-Aug-1993.)

Theoremeqcoms 2404 Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 5-Aug-1993.)

Theoremeqcomi 2405 Inference from commutative law for class equality. (Contributed by NM, 5-Aug-1993.)

Theoremeqcomd 2406 Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.)

Theoremeqeq1 2407 Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.)

Theoremeqeq1i 2408 Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.)

Theoremeqeq1d 2409 Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.)

Theoremeqeq2 2410 Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.)

Theoremeqeq2i 2411 Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.)

Theoremeqeq2d 2412 Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.)

Theoremeqeq12 2413 Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.)

Theoremeqeq12i 2414 A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremeqeq12d 2415 A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremeqeqan12d 2416 A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremeqeqan12rd 2417 A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.)

Theoremeqtr 2418 Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.)

Theoremeqtr2 2419 A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremeqtr3 2420 A transitive law for class equality. (Contributed by NM, 20-May-2005.)

Theoremeqtri 2421 An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)

Theoremeqtr2i 2422 An equality transitivity inference. (Contributed by NM, 21-Feb-1995.)

Theoremeqtr3i 2423 An equality transitivity inference. (Contributed by NM, 6-May-1994.)

Theoremeqtr4i 2424 An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)

Theorem3eqtri 2425 An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.)

Theorem3eqtrri 2426 An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem3eqtr2i 2427 An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.)

Theorem3eqtr2ri 2428 An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem3eqtr3i 2429 An inference from three chained equalities. (Contributed by NM, 6-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem3eqtr3ri 2430 An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.)

Theorem3eqtr4i 2431 An inference from three chained equalities. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem3eqtr4ri 2432 An inference from three chained equalities. (Contributed by NM, 2-Sep-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremeqtrd 2433 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)

Theoremeqtr2d 2434 An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.)

Theoremeqtr3d 2435 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)

Theoremeqtr4d 2436 An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)

Theorem3eqtrd 2437 A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.)

Theorem3eqtrrd 2438 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem3eqtr2d 2439 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)

Theorem3eqtr2rd 2440 A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)

Theorem3eqtr3d 2441 A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem3eqtr3rd 2442 A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.)

Theorem3eqtr4d 2443 A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theorem3eqtr4rd 2444 A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995.)

Theoremsyl5eq 2445 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)

Theoremsyl5req 2446 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)

Theoremsyl5eqr 2447 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)

Theoremsyl5reqr 2448 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)

Theoremsyl6eq 2449 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)

Theoremsyl6req 2450 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)

Theoremsyl6eqr 2451 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)

Theoremsyl6reqr 2452 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)

Theoremsylan9eq 2453 An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremsylan9req 2454 An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)

Theoremsylan9eqr 2455 An equality transitivity deduction. (Contributed by NM, 8-May-1994.)

Theorem3eqtr3g 2456 A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)

Theorem3eqtr3a 2457 A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.)

Theorem3eqtr4g 2458 A chained equality inference, useful for converting to definitions. (Contributed by NM, 5-Aug-1993.)

Theorem3eqtr4a 2459 A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremeq2tri 2460 A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)

Theoremeleq1 2461 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq2 2462 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq12 2463 Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)

Theoremeleq1i 2464 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq2i 2465 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq12i 2466 Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)

Theoremeleq1d 2467 Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)

Theoremeleq2d 2468 Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.)

Theoremeleq12d 2469 Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)

Theoremeleq1a 2470 A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)

Theoremeqeltri 2471 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Theoremeqeltrri 2472 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Theoremeleqtri 2473 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Theoremeleqtrri 2474 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Theoremeqeltrd 2475 Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremeqeltrrd 2476 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)

Theoremeleqtrd 2477 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)

Theoremeleqtrrd 2478 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)

Theorem3eltr3i 2479 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr4i 2480 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr3d 2481 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr4d 2482 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr3g 2483 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theorem3eltr4g 2484 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theoremsyl5eqel 2485 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl5eqelr 2486 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl5eleq 2487 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl5eleqr 2488 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eqel 2489 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eqelr 2490 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eleq 2491 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Theoremsyl6eleqr 2492 A membership and equality inference. (Contributed by NM, 24-Apr-2005.)

Theoremeleq2s 2493 Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremeqneltrd 2494 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 2495 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 2496 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 2497 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 2498* Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 5-Aug-1993.)

Theoremnelneq 2499 A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)

Theoremnelneq2 2500 A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)

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