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Theorem List for New Foundations Explorer - 2501-2600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnfel2 2501* Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
xB       x A B

Theoremnfcrd 2502* Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
(φxA)       (φ → Ⅎx y A)

Theoremnfeqd 2503 Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)
(φxA)    &   (φxB)       (φ → Ⅎx A = B)

Theoremnfeld 2504 Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)
(φxA)    &   (φxB)       (φ → Ⅎx A B)

Theoremdrnfc1 2505 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)
(x x = yA = B)       (x x = y → (xAyB))

Theoremdrnfc2 2506 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)
(x x = yA = B)       (x x = y → (zAzB))

Theoremnfabd2 2507 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
yφ    &   ((φ ¬ x x = y) → Ⅎxψ)       (φx{y ψ})

Theoremnfabd 2508 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
yφ    &   (φ → Ⅎxψ)       (φx{y ψ})

Theoremdvelimdc 2509 Deduction form of dvelimc 2510. (Contributed by Mario Carneiro, 8-Oct-2016.)
xφ    &   zφ    &   (φxA)    &   (φzB)    &   (φ → (z = yA = B))       (φ → (¬ x x = yxB))

Theoremdvelimc 2510 Version of dvelim 2016 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)
xA    &   zB    &   (z = yA = B)       x x = yxB)

Theoremnfcvf 2511 If x and y are distinct, then x is not free in y. (Contributed by Mario Carneiro, 8-Oct-2016.)
x x = yxy)

Theoremnfcvf2 2512 If x and y are distinct, then y is not free in x. (Contributed by Mario Carneiro, 5-Dec-2016.)
x x = yyx)

Theoremcleqf 2513 Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
xA    &   xB       (A = Bx(x Ax B))

Theoremabid2f 2514 A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
xA       {x x A} = A

Theoremsbabel 2515* Theorem to move a substitution in and out of a class abstraction. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.)
xA       ([y / x]{z φ} A ↔ {z [y / x]φ} A)

2.1.4  Negated equality and membership

Syntaxwne 2516 Extend wff notation to include inequality.
wff AB

Syntaxwnel 2517 Extend wff notation to include negated membership.
wff A B

Definitiondf-ne 2518 Define inequality. (Contributed by NM, 5-Aug-1993.)
(AB ↔ ¬ A = B)

Definitiondf-nel 2519 Define negated membership. (Contributed by NM, 7-Aug-1994.)
(A B ↔ ¬ A B)

Theoremnne 2520 Negation of inequality. (Contributed by NM, 9-Jun-2006.)
ABA = B)

Theoremneirr 2521 No class is unequal to itself. (Contributed by Stefan O'Rear, 1-Jan-2015.)
¬ AA

Theoremexmidne 2522 Excluded middle with equality and inequality. (Contributed by NM, 3-Feb-2012.)
(A = B AB)

Theoremnonconne 2523 Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.)
¬ (A = B AB)

Theoremneeq1 2524 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
(A = B → (ACBC))

Theoremneeq2 2525 Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
(A = B → (CACB))

Theoremneeq1i 2526 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
A = B       (ACBC)

Theoremneeq2i 2527 Inference for inequality. (Contributed by NM, 29-Apr-2005.)
A = B       (CACB)

Theoremneeq12i 2528 Inference for inequality. (Contributed by NM, 24-Jul-2012.)
A = B    &   C = D       (ACBD)

Theoremneeq1d 2529 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
(φA = B)       (φ → (ACBC))

Theoremneeq2d 2530 Deduction for inequality. (Contributed by NM, 25-Oct-1999.)
(φA = B)       (φ → (CACB))

Theoremneeq12d 2531 Deduction for inequality. (Contributed by NM, 24-Jul-2012.)
(φA = B)    &   (φC = D)       (φ → (ACBD))

Theoremneneqd 2532 Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(φAB)       (φ → ¬ A = B)

Theoremeqnetri 2533 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
A = B    &   BC       AC

Theoremeqnetrd 2534 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(φA = B)    &   (φBC)       (φAC)

Theoremeqnetrri 2535 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
A = B    &   AC       BC

Theoremeqnetrrd 2536 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(φA = B)    &   (φAC)       (φBC)

Theoremneeqtri 2537 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
AB    &   B = C       AC

Theoremneeqtrd 2538 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(φAB)    &   (φB = C)       (φAC)

Theoremneeqtrri 2539 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
AB    &   C = B       AC

Theoremneeqtrrd 2540 Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.)
(φAB)    &   (φC = B)       (φAC)

Theoremsyl5eqner 2541 B chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.)
B = A    &   (φBC)       (φAC)

Theorem3netr3d 2542 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
(φAB)    &   (φA = C)    &   (φB = D)       (φCD)

Theorem3netr4d 2543 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
(φAB)    &   (φC = A)    &   (φD = B)       (φCD)

Theorem3netr3g 2544 Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
(φAB)    &   A = C    &   B = D       (φCD)

Theorem3netr4g 2545 Substitution of equality into both sides of an inequality. (Contributed by NM, 14-Jun-2012.)
(φAB)    &   C = A    &   D = B       (φCD)

Theoremnecon3abii 2546 Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)
(A = Bφ)       (AB ↔ ¬ φ)

Theoremnecon3bbii 2547 Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)
(φA = B)       φAB)

Theoremnecon3bii 2548 Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)
(A = BC = D)       (ABCD)

Theoremnecon3abid 2549 Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)
(φ → (A = Bψ))       (φ → (AB ↔ ¬ ψ))

Theoremnecon3bbid 2550 Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)
(φ → (ψA = B))       (φ → (¬ ψAB))

Theoremnecon3bid 2551 Deduction from equality to inequality. (Contributed by NM, 23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φ → (A = BC = D))       (φ → (ABCD))

Theoremnecon3ad 2552 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φ → (ψA = B))       (φ → (AB → ¬ ψ))

Theoremnecon3bd 2553 Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φ → (A = Bψ))       (φ → (¬ ψAB))

Theoremnecon3d 2554 Contrapositive law deduction for inequality. (Contributed by NM, 10-Jun-2006.)
(φ → (A = BC = D))       (φ → (CDAB))

Theoremnecon3i 2555 Contrapositive inference for inequality. (Contributed by NM, 9-Aug-2006.)
(A = BC = D)       (CDAB)

Theoremnecon3ai 2556 Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φA = B)       (AB → ¬ φ)

Theoremnecon3bi 2557 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(A = Bφ)       φAB)

Theoremnecon1ai 2558 Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007.)
φA = B)       (ABφ)

Theoremnecon1bi 2559 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(ABφ)       φA = B)

Theoremnecon1i 2560 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
(ABC = D)       (CDA = B)

Theoremnecon2ai 2561 Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(A = B → ¬ φ)       (φAB)

Theoremnecon2bi 2562 Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)
(φAB)       (A = B → ¬ φ)

Theoremnecon2i 2563 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
(A = BCD)       (C = DAB)

Theoremnecon2ad 2564 Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φ → (A = B → ¬ ψ))       (φ → (ψAB))

Theoremnecon2bd 2565 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
(φ → (ψAB))       (φ → (A = B → ¬ ψ))

Theoremnecon2d 2566 Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
(φ → (A = BCD))       (φ → (C = DAB))

Theoremnecon1abii 2567 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.)
φA = B)       (ABφ)

Theoremnecon1bbii 2568 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.)
(ABφ)       φA = B)

Theoremnecon1abid 2569 Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.)
(φ → (¬ ψA = B))       (φ → (ABψ))

Theoremnecon1bbid 2570 Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008.)
(φ → (ABψ))       (φ → (¬ ψA = B))

Theoremnecon2abii 2571 Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007.)
(A = B ↔ ¬ φ)       (φAB)

Theoremnecon2bbii 2572 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
(φAB)       (A = B ↔ ¬ φ)

Theoremnecon2abid 2573 Contrapositive deduction for inequality. (Contributed by NM, 18-Jul-2007.)
(φ → (A = B ↔ ¬ ψ))       (φ → (ψAB))

Theoremnecon2bbid 2574 Contrapositive deduction for inequality. (Contributed by NM, 13-Apr-2007.)
(φ → (ψAB))       (φ → (A = B ↔ ¬ ψ))

Theoremnecon4ai 2575 Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(AB → ¬ φ)       (φA = B)

Theoremnecon4i 2576 Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(ABCD)       (C = DA = B)

Theoremnecon4ad 2577 Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φ → (AB → ¬ ψ))       (φ → (ψA = B))

Theoremnecon4bd 2578 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φ → (¬ ψAB))       (φ → (A = Bψ))

Theoremnecon4d 2579 Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φ → (ABCD))       (φ → (C = DA = B))

Theoremnecon4abid 2580 Contrapositive law deduction for inequality. (Contributed by NM, 11-Jan-2008.)
(φ → (AB ↔ ¬ ψ))       (φ → (A = Bψ))

Theoremnecon4bbid 2581 Contrapositive law deduction for inequality. (Contributed by NM, 9-May-2012.)
(φ → (¬ ψAB))       (φ → (ψA = B))

Theoremnecon4bid 2582 Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.)
(φ → (ABCD))       (φ → (A = BC = D))

Theoremnecon1ad 2583 Contrapositive deduction for inequality. (Contributed by NM, 2-Apr-2007.)
(φ → (¬ ψA = B))       (φ → (ABψ))

Theoremnecon1bd 2584 Contrapositive deduction for inequality. (Contributed by NM, 21-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φ → (ABψ))       (φ → (¬ ψA = B))

Theoremnecon1d 2585 Contrapositive law deduction for inequality. (Contributed by NM, 28-Dec-2008.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φ → (ABC = D))       (φ → (CDA = B))

Theoremneneqad 2586 If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2532. One-way deduction form of df-ne 2518. (Contributed by David Moews, 28-Feb-2017.)
(φ → ¬ A = B)       (φAB)

Theoremnebi 2587 Contraposition law for inequality. (Contributed by NM, 28-Dec-2008.)
((A = BC = D) ↔ (ABCD))

Theorempm13.18 2588 Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
((A = B AC) → BC)

Theorempm13.181 2589 Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
((A = B BC) → AC)

Theorempm2.21ddne 2590 A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)
(φA = B)    &   (φAB)       (φψ)

Theorempm2.61ne 2591 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(A = B → (ψχ))    &   ((φ AB) → ψ)    &   (φχ)       (φψ)

Theorempm2.61ine 2592 Inference eliminating an inequality in an antecedent. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(A = Bφ)    &   (ABφ)       φ

Theorempm2.61dne 2593 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(φ → (A = Bψ))    &   (φ → (ABψ))       (φψ)

Theorempm2.61dane 2594 Deduction eliminating an inequality in an antecedent. (Contributed by NM, 30-Nov-2011.)
((φ A = B) → ψ)    &   ((φ AB) → ψ)       (φψ)

Theorempm2.61da2ne 2595 Deduction eliminating two inequalities in an antecedent. (Contributed by NM, 29-May-2013.)
((φ A = B) → ψ)    &   ((φ C = D) → ψ)    &   ((φ (AB CD)) → ψ)       (φψ)

Theorempm2.61da3ne 2596 Deduction eliminating three inequalities in an antecedent. (Contributed by NM, 15-Jun-2013.)
((φ A = B) → ψ)    &   ((φ C = D) → ψ)    &   ((φ E = F) → ψ)    &   ((φ (AB CD EF)) → ψ)       (φψ)

Theoremnecom 2597 Commutation of inequality. (Contributed by NM, 14-May-1999.)
(ABBA)

Theoremnecomi 2598 Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)
AB       BA

Theoremnecomd 2599 Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)
(φAB)       (φBA)

Theoremneor 2600 Logical OR with an equality. (Contributed by NM, 29-Apr-2007.)
((A = B ψ) ↔ (ABψ))

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