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Theorem List for New Foundations Explorer - 2601-2700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremneanior 2601 A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
((AB CD) ↔ ¬ (A = B C = D))

Theoremne3anior 2602 A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.)
((AB CD EF) ↔ ¬ (A = B C = D E = F))

Theoremneorian 2603 A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
((AB CD) ↔ ¬ (A = B C = D))

Theoremnemtbir 2604 An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
AB    &   (φA = B)        ¬ φ

Theoremnelne1 2605 Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)
((A B ¬ A C) → BC)

Theoremnelne2 2606 Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.)
((A C ¬ B C) → AB)

Theoremneleq1 2607 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
(A = B → (A CB C))

Theoremneleq2 2608 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
(A = B → (C AC B))

Theoremneleq12d 2609 Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.)
(φA = B)    &   (φC = D)       (φ → (A CB D))

Theoremnfne 2610 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
xA    &   xB       x AB

Theoremnfnel 2611 Bound-variable hypothesis builder for inequality. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
xA    &   xB       x A B

Theoremnfned 2612 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
(φxA)    &   (φxB)       (φ → Ⅎx AB)

Theoremnfneld 2613 Bound-variable hypothesis builder for inequality. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
(φxA)    &   (φxB)       (φ → Ⅎx A B)

2.1.5  Restricted quantification

Syntaxwral 2614 Extend wff notation to include restricted universal quantification.
wff x A φ

Syntaxwrex 2615 Extend wff notation to include restricted existential quantification.
wff x A φ

Syntaxwreu 2616 Extend wff notation to include restricted existential uniqueness.
wff ∃!x A φ

Syntaxwrmo 2617 Extend wff notation to include restricted "at most one."
wff ∃*x A φ

Syntaxcrab 2618 Extend class notation to include the restricted class abstraction (class builder).
class {x A φ}

Definitiondf-ral 2619 Define restricted universal quantification. Special case of Definition 4.15(3) of [TakeutiZaring] p. 22. (Contributed by NM, 19-Aug-1993.)
(x A φx(x Aφ))

Definitiondf-rex 2620 Define restricted existential quantification. Special case of Definition 4.15(4) of [TakeutiZaring] p. 22. (Contributed by NM, 30-Aug-1993.)
(x A φx(x A φ))

Definitiondf-reu 2621 Define restricted existential uniqueness. (Contributed by NM, 22-Nov-1994.)
(∃!x A φ∃!x(x A φ))

Definitiondf-rmo 2622 Define restricted "at most one". (Contributed by NM, 16-Jun-2017.)
(∃*x A φ∃*x(x A φ))

Definitiondf-rab 2623 Define a restricted class abstraction (class builder), which is the class of all x in A such that φ is true. Definition of [TakeutiZaring] p. 20. (Contributed by NM, 22-Nov-1994.)
{x A φ} = {x (x A φ)}

Theoremralnex 2624 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
(x A ¬ φ ↔ ¬ x A φ)

Theoremrexnal 2625 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
(x A ¬ φ ↔ ¬ x A φ)

Theoremdfral2 2626 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
(x A φ ↔ ¬ x A ¬ φ)

Theoremdfrex2 2627 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
(x A φ ↔ ¬ x A ¬ φ)

Theoremralbida 2628 Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)
xφ    &   ((φ x A) → (ψχ))       (φ → (x A ψx A χ))

Theoremrexbida 2629 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)
xφ    &   ((φ x A) → (ψχ))       (φ → (x A ψx A χ))

Theoremralbidva 2630* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 4-Mar-1997.)
((φ x A) → (ψχ))       (φ → (x A ψx A χ))

Theoremrexbidva 2631* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 9-Mar-1997.)
((φ x A) → (ψχ))       (φ → (x A ψx A χ))

Theoremralbid 2632 Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)
xφ    &   (φ → (ψχ))       (φ → (x A ψx A χ))

Theoremrexbid 2633 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)
xφ    &   (φ → (ψχ))       (φ → (x A ψx A χ))

Theoremralbidv 2634* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 20-Nov-1994.)
(φ → (ψχ))       (φ → (x A ψx A χ))

Theoremrexbidv 2635* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 20-Nov-1994.)
(φ → (ψχ))       (φ → (x A ψx A χ))

Theoremralbidv2 2636* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Apr-1997.)
(φ → ((x Aψ) ↔ (x Bχ)))       (φ → (x A ψx B χ))

Theoremrexbidv2 2637* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 22-May-1999.)
(φ → ((x A ψ) ↔ (x B χ)))       (φ → (x A ψx B χ))

Theoremralbii 2638 Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario Carneiro, 17-Oct-2016.)
(φψ)       (x A φx A ψ)

Theoremrexbii 2639 Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario Carneiro, 17-Oct-2016.)
(φψ)       (x A φx A ψ)

Theorem2ralbii 2640 Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
(φψ)       (x A y B φx A y B ψ)

Theorem2rexbii 2641 Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 11-Nov-1995.)
(φψ)       (x A y B φx A y B ψ)

Theoremralbii2 2642 Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005.)
((x Aφ) ↔ (x Bψ))       (x A φx B ψ)

Theoremrexbii2 2643 Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.)
((x A φ) ↔ (x B ψ))       (x A φx B ψ)

Theoremraleqbii 2644 Equality deduction for restricted universal quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
A = B    &   (ψχ)       (x A ψx B χ)

Theoremrexeqbii 2645 Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
A = B    &   (ψχ)       (x A ψx B χ)

Theoremralbiia 2646 Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000.)
(x A → (φψ))       (x A φx A ψ)

Theoremrexbiia 2647 Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 26-Oct-1999.)
(x A → (φψ))       (x A φx A ψ)

Theorem2rexbiia 2648* Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
((x A y B) → (φψ))       (x A y B φx A y B ψ)

Theoremr2alf 2649* Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
yA       (x A y B φxy((x A y B) → φ))

Theoremr2exf 2650* Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
yA       (x A y B φxy((x A y B) φ))

Theoremr2al 2651* Double restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
(x A y B φxy((x A y B) → φ))

Theoremr2ex 2652* Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.)
(x A y B φxy((x A y B) φ))

Theorem2ralbida 2653* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 24-Feb-2004.)
xφ    &   yφ    &   ((φ (x A y B)) → (ψχ))       (φ → (x A y B ψx A y B χ))

Theorem2ralbidva 2654* Formula-building rule for restricted universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)
((φ (x A y B)) → (ψχ))       (φ → (x A y B ψx A y B χ))

Theorem2rexbidva 2655* Formula-building rule for restricted existential quantifiers (deduction rule). (Contributed by NM, 15-Dec-2004.)
((φ (x A y B)) → (ψχ))       (φ → (x A y B ψx A y B χ))

Theorem2ralbidv 2656* Formula-building rule for restricted universal quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon Jaroszewicz, 16-Mar-2007.)
(φ → (ψχ))       (φ → (x A y B ψx A y B χ))

Theorem2rexbidv 2657* Formula-building rule for restricted existential quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.)
(φ → (ψχ))       (φ → (x A y B ψx A y B χ))

Theoremrexralbidv 2658* Formula-building rule for restricted quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.)
(φ → (ψχ))       (φ → (x A y B ψx A y B χ))

Theoremralinexa 2659 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
(x A (φ → ¬ ψ) ↔ ¬ x A (φ ψ))

Theoremrexanali 2660 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
(x A (φ ¬ ψ) ↔ ¬ x A (φψ))

Theoremrisset 2661* Two ways to say "A belongs to B." (Contributed by NM, 22-Nov-1994.)
(A Bx B x = A)

Theoremhbral 2662 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy, 13-Dec-2009.)
(y Ax y A)    &   (φxφ)       (y A φxy A φ)

Theoremhbra1 2663 x is not free in x Aφ. (Contributed by NM, 18-Oct-1996.)
(x A φxx A φ)

Theoremnfra1 2664 x is not free in x Aφ. (Contributed by NM, 18-Oct-1996.) (Revised by Mario Carneiro, 7-Oct-2016.)
xx A φ

Theoremnfrald 2665 Deduction version of nfral 2667. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)
yφ    &   (φxA)    &   (φ → Ⅎxψ)       (φ → Ⅎxy A ψ)

Theoremnfrexd 2666 Deduction version of nfrex 2669. (Contributed by Mario Carneiro, 14-Oct-2016.)
yφ    &   (φxA)    &   (φ → Ⅎxψ)       (φ → Ⅎxy A ψ)

Theoremnfral 2667 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.)
xA    &   xφ       xy A φ

Theoremnfra2 2668* Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD in set.mm. Contributed by Alan Sare 31-Dec-2011. (Contributed by NM, 31-Dec-2011.)
yx A y B φ

Theoremnfrex 2669 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.)
xA    &   xφ       xy A φ

Theoremnfre1 2670 x is not free in x Aφ. (Contributed by NM, 19-Mar-1997.) (Revised by Mario Carneiro, 7-Oct-2016.)
xx A φ

Theoremr3al 2671* Triple restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
(x A y B z C φxyz((x A y B z C) → φ))

Theoremalral 2672 Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
(xφx A φ)

Theoremrexex 2673 Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
(x A φxφ)

Theoremrsp 2674 Restricted specialization. (Contributed by NM, 17-Oct-1996.)
(x A φ → (x Aφ))

Theoremrspe 2675 Restricted specialization. (Contributed by NM, 12-Oct-1999.)
((x A φ) → x A φ)

Theoremrsp2 2676 Restricted specialization. (Contributed by NM, 11-Feb-1997.)
(x A y B φ → ((x A y B) → φ))

Theoremrsp2e 2677 Restricted specialization. (Contributed by FL, 4-Jun-2012.)
((x A y B φ) → x A y B φ)

Theoremrspec 2678 Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
x A φ       (x Aφ)

Theoremrgen 2679 Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
(x Aφ)       x A φ

Theoremrgen2a 2680* Generalization rule for restricted quantification. Note that x and y needn't be distinct (and illustrates the use of dvelim 2016). (Contributed by NM, 23-Nov-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.
((x A y A) → φ)       x A y A φ

Theoremrgenw 2681 Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
φ       x A φ

Theoremrgen2w 2682 Generalization rule for restricted quantification. Note that x and y needn't be distinct. (Contributed by NM, 18-Jun-2014.)
φ       x A y B φ

Theoremmprg 2683 Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)
(x A φψ)    &   (x Aφ)       ψ

Theoremmprgbir 2684 Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.)
(φx A ψ)    &   (x Aψ)       φ

Theoremralim 2685 Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.)
(x A (φψ) → (x A φx A ψ))

Theoremralimi2 2686 Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)
((x Aφ) → (x Bψ))       (x A φx B ψ)

Theoremralimia 2687 Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)
(x A → (φψ))       (x A φx A ψ)

Theoremralimiaa 2688 Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.)
((x A φ) → ψ)       (x A φx A ψ)

Theoremralimi 2689 Inference quantifying both antecedent and consequent, with strong hypothesis. (Contributed by NM, 4-Mar-1997.)
(φψ)       (x A φx A ψ)

Theoremral2imi 2690 Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.)
(φ → (ψχ))       (x A φ → (x A ψx A χ))

Theoremralimdaa 2691 Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.)
xφ    &   ((φ x A) → (ψχ))       (φ → (x A ψx A χ))

Theoremralimdva 2692* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)
((φ x A) → (ψχ))       (φ → (x A ψx A χ))

Theoremralimdv 2693* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 8-Oct-2003.)
(φ → (ψχ))       (φ → (x A ψx A χ))

Theoremralimdv2 2694* Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005.)
(φ → ((x Aψ) → (x Bχ)))       (φ → (x A ψx B χ))

Theoremralrimi 2695 Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.)
xφ    &   (φ → (x Aψ))       (φx A ψ)

Theoremralrimiv 2696* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.)
(φ → (x Aψ))       (φx A ψ)

Theoremralrimiva 2697* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Jan-2006.)
((φ x A) → ψ)       (φx A ψ)

Theoremralrimivw 2698* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
(φψ)       (φx A ψ)

Theoremr19.21t 2699 Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed theorem version). (Contributed by NM, 1-Mar-2008.)
(Ⅎxφ → (x A (φψ) ↔ (φx A ψ)))

Theoremr19.21 2700 Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 30-Mar-2011.)
xφ       (x A (φψ) ↔ (φx A ψ))

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