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

Theoremsyl6eq 2401 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
(φA = B)    &   B = C       (φA = C)

Theoremsyl6req 2402 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
(φA = B)    &   B = C       (φC = A)

Theoremsyl6eqr 2403 An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
(φA = B)    &   C = B       (φA = C)

Theoremsyl6reqr 2404 An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
(φA = B)    &   C = B       (φC = A)

Theoremsylan9eq 2405 An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φA = B)    &   (ψB = C)       ((φ ψ) → A = C)

Theoremsylan9req 2406 An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)
(φB = A)    &   (ψB = C)       ((φ ψ) → A = C)

Theoremsylan9eqr 2407 An equality transitivity deduction. (Contributed by NM, 8-May-1994.)
(φA = B)    &   (ψB = C)       ((ψ φ) → A = C)

Theorem3eqtr3g 2408 A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)
(φA = B)    &   A = C    &   B = D       (φC = D)

Theorem3eqtr3a 2409 A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.)
A = B    &   (φA = C)    &   (φB = D)       (φC = D)

Theorem3eqtr4g 2410 A chained equality inference, useful for converting to definitions. (Contributed by NM, 5-Aug-1993.)
(φA = B)    &   C = A    &   D = B       (φC = D)

Theorem3eqtr4a 2411 A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
A = B    &   (φC = A)    &   (φD = B)       (φC = D)

Theoremeq2tri 2412 A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)
(A = CD = F)    &   (B = DC = G)       ((A = C B = F) ↔ (B = D A = G))

Theoremeleq1 2413 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
(A = B → (A CB C))

Theoremeleq2 2414 Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
(A = B → (C AC B))

Theoremeleq12 2415 Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.)
((A = B C = D) → (A CB D))

Theoremeleq1i 2416 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
A = B       (A CB C)

Theoremeleq2i 2417 Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
A = B       (C AC B)

Theoremeleq12i 2418 Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
A = B    &   C = D       (A CB D)

Theoremeleq1d 2419 Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
(φA = B)       (φ → (A CB C))

Theoremeleq2d 2420 Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.)
(φA = B)       (φ → (C AC B))

Theoremeleq12d 2421 Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
(φA = B)    &   (φC = D)       (φ → (A CB D))

Theoremeleq1a 2422 A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)
(A B → (C = AC B))

Theoremeqeltri 2423 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
A = B    &   B C       A C

Theoremeqeltrri 2424 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
A = B    &   A C       B C

Theoremeleqtri 2425 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
A B    &   B = C       A C

Theoremeleqtrri 2426 Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
A B    &   C = B       A C

Theoremeqeltrd 2427 Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)
(φA = B)    &   (φB C)       (φA C)

Theoremeqeltrrd 2428 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
(φA = B)    &   (φA C)       (φB C)

Theoremeleqtrd 2429 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
(φA B)    &   (φB = C)       (φA C)

Theoremeleqtrrd 2430 Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
(φA B)    &   (φC = B)       (φA C)

Theorem3eltr3i 2431 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
A B    &   A = C    &   B = D       C D

Theorem3eltr4i 2432 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
A B    &   C = A    &   D = B       C D

Theorem3eltr3d 2433 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(φA B)    &   (φA = C)    &   (φB = D)       (φC D)

Theorem3eltr4d 2434 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(φA B)    &   (φC = A)    &   (φD = B)       (φC D)

Theorem3eltr3g 2435 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(φA B)    &   A = C    &   B = D       (φC D)

Theorem3eltr4g 2436 Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
(φA B)    &   C = A    &   D = B       (φC D)

Theoremsyl5eqel 2437 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
A = B    &   (φB C)       (φA C)

Theoremsyl5eqelr 2438 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
B = A    &   (φB C)       (φA C)

Theoremsyl5eleq 2439 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
A B    &   (φB = C)       (φA C)

Theoremsyl5eleqr 2440 B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
A B    &   (φC = B)       (φA C)

Theoremsyl6eqel 2441 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(φA = B)    &   B C       (φA C)

Theoremsyl6eqelr 2442 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(φB = A)    &   B C       (φA C)

Theoremsyl6eleq 2443 A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
(φA B)    &   B = C       (φA C)

Theoremsyl6eleqr 2444 A membership and equality inference. (Contributed by NM, 24-Apr-2005.)
(φA B)    &   C = B       (φA C)

Theoremeleq2s 2445 Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(A Bφ)    &   C = B       (A Cφ)

Theoremeqneltrd 2446 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.)
(φA = B)    &   (φ → ¬ B C)       (φ → ¬ A C)

Theoremeqneltrrd 2447 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.)
(φA = B)    &   (φ → ¬ A C)       (φ → ¬ B C)

Theoremneleqtrd 2448 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.)
(φ → ¬ C A)    &   (φA = B)       (φ → ¬ C B)

Theoremneleqtrrd 2449 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.)
(φ → ¬ C B)    &   (φA = B)       (φ → ¬ C A)

Theoremcleqh 2450* Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 5-Aug-1993.)
(y Ax y A)    &   (y Bx y B)       (A = Bx(x Ax B))

Theoremnelneq 2451 A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)
((A C ¬ B C) → ¬ A = B)

Theoremnelneq2 2452 A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
((A B ¬ A C) → ¬ B = C)

Theoremeqsb3lem 2453* Lemma for eqsb3 2454. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([y / x]x = Ay = A)

Theoremeqsb3 2454* Substitution applied to an atomic wff (class version of equsb3 2102). (Contributed by Rodolfo Medina, 28-Apr-2010.)
([y / x]x = Ay = A)

Theoremclelsb3 2455* Substitution applied to an atomic wff (class version of elsb3 2103). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
([y / x]x Ay A)

Theoremhbxfreq 2456 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1568 for equivalence version. (Contributed by NM, 21-Aug-2007.)
A = B    &   (y Bx y B)       (y Ax y A)

Theoremhblem 2457* Change the free variable of a hypothesis builder. Lemma for nfcrii 2482. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)
(y Ax y A)       (z Ax z A)

Theoremabeq2 2458* 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 2463 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 φ (that has a free variable x) to a theorem with a class variable A, we substitute x A for φ throughout and simplify, where A is a new class variable not already in the wff. An example is the conversion of zfauscl in set.mm to inex1 in set.mm (look at the instance of zfauscl that occurs in the proof of inex1 ). Conversely, to convert a theorem with a class variable A to one with φ, we substitute {x φ} for A throughout and simplify, where x and φ are new setvar and wff variables not already in the wff. An example is cp in set.mm , which derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 in set.mm. 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.)

(A = {x φ} ↔ x(x Aφ))

Theoremabeq1 2459* Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)
({x φ} = Ax(φx A))

Theoremabeq2i 2460 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 3-Apr-1996.)
A = {x φ}       (x Aφ)

Theoremabeq1i 2461 Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.)
{x φ} = A       (φx A)

Theoremabeq2d 2462 Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)
(φA = {x ψ})       (φ → (x Aψ))

Theoremabbi 2463 Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.)
(x(φψ) ↔ {x φ} = {x ψ})

Theoremabbi2i 2464* Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 5-Aug-1993.)
(x Aφ)       A = {x φ}

Theoremabbii 2465 Equivalent wff's yield equal class abstractions (inference rule). (Contributed by NM, 5-Aug-1993.)
(φψ)       {x φ} = {x ψ}

Theoremabbid 2466 Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
xφ    &   (φ → (ψχ))       (φ → {x ψ} = {x χ})

Theoremabbidv 2467* Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 10-Aug-1993.)
(φ → (ψχ))       (φ → {x ψ} = {x χ})

Theoremabbi2dv 2468* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
(φ → (x Aψ))       (φA = {x ψ})

Theoremabbi1dv 2469* Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
(φ → (ψx A))       (φ → {x ψ} = A)

Theoremabid2 2470* A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)
{x x A} = A

Theoremcbvab 2471 Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)
yφ    &   xψ    &   (x = y → (φψ))       {x φ} = {y ψ}

Theoremcbvabv 2472* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)
(x = y → (φψ))       {x φ} = {y ψ}

Theoremclelab 2473* Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)
(A {x φ} ↔ x(x = A φ))

Theoremclabel 2474* Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)
({x φ} Ay(y A x(x yφ)))

Theoremsbab 2475* The right-hand side of the second equality is a way of representing proper substitution of y for x into a class variable. (Contributed by NM, 14-Sep-2003.)
(x = yA = {z [y / x]z A})

2.1.3  Class form not-free predicate

Syntaxwnfc 2476 Extend wff definition to include the not-free predicate for classes.
wff xA

Theoremnfcjust 2477* Justification theorem for df-nfc 2478. (Contributed by Mario Carneiro, 13-Oct-2016.)
(yx y Azx z A)

Definitiondf-nfc 2478* Define the not-free predicate for classes. This is read "x is not free in A". Not-free means that the value of x cannot affect the value of A, e.g., any occurrence of x in A 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 1545 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)
(xAyx y A)

Theoremnfci 2479* Deduce that a class A does not have x free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
x y A       xA

Theoremnfcii 2480* Deduce that a class A does not have x free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
(y Ax y A)       xA

Theoremnfcr 2481* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
(xA → Ⅎx y A)

Theoremnfcrii 2482* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
xA       (y Ax y A)

Theoremnfcri 2483* Consequence of the not-free predicate. (Note that unlike nfcr 2481, this does not require y and A to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.)
xA       x y A

Theoremnfcd 2484* Deduce that a class A does not have x free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
yφ    &   (φ → Ⅎx y A)       (φxA)

Theoremnfceqi 2485 Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
A = B       (xAxB)

Theoremnfcxfr 2486 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
A = B    &   xB       xA

Theoremnfcxfrd 2487 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
A = B    &   (φxB)       (φxA)

Theoremnfceqdf 2488 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)
xφ    &   (φA = B)       (φ → (xAxB))

Theoremnfcv 2489* If x is disjoint from A, then x is not free in A. (Contributed by Mario Carneiro, 11-Aug-2016.)
xA

Theoremnfcvd 2490* If x is disjoint from A, then x is not free in A. (Contributed by Mario Carneiro, 7-Oct-2016.)
(φxA)

Theoremnfab1 2491 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
x{x φ}

Theoremnfnfc1 2492 x is bound in xA. (Contributed by Mario Carneiro, 11-Aug-2016.)
xxA

Theoremnfab 2493 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
xφ       x{y φ}

Theoremnfaba1 2494 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)
x{y xφ}

Theoremnfnfc 2495 Hypothesis builder for yA. (Contributed by Mario Carneiro, 11-Aug-2016.)
xA       xyA

Theoremnfeq 2496 Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
xA    &   xB       x A = B

Theoremnfel 2497 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
xA    &   xB       x A B

Theoremnfeq1 2498* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
xA       x A = B

Theoremnfel1 2499* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
xA       x A B

Theoremnfeq2 2500* Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
xB       x A = B

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