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Theorem List for Metamath Proof Explorer - 2701-2800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremneleq12d 2701 Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.)

Theoremnfnel 2702 Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfneld 2703 Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnnel 2704 Negation of negated membership, analogous to nne 2605. (Contributed by Alexander van der Vekens, 18-Jan-2018.)

2.1.5  Restricted quantification

Syntaxwral 2705 Extend wff notation to include restricted universal quantification.

Syntaxwrex 2706 Extend wff notation to include restricted existential quantification.

Syntaxwreu 2707 Extend wff notation to include restricted existential uniqueness.

Syntaxwrmo 2708 Extend wff notation to include restricted "at most one."

Syntaxcrab 2709 Extend class notation to include the restricted class abstraction (class builder).

Definitiondf-ral 2710 Define restricted universal quantification. Special case of Definition 4.15(3) of [TakeutiZaring] p. 22. (Contributed by NM, 19-Aug-1993.)

Definitiondf-rex 2711 Define restricted existential quantification. Special case of Definition 4.15(4) of [TakeutiZaring] p. 22. (Contributed by NM, 30-Aug-1993.)

Definitiondf-reu 2712 Define restricted existential uniqueness. (Contributed by NM, 22-Nov-1994.)

Definitiondf-rmo 2713 Define restricted "at most one". (Contributed by NM, 16-Jun-2017.)

Definitiondf-rab 2714 Define a restricted class abstraction (class builder), which is the class of all in such that is true. Definition of [TakeutiZaring] p. 20. (Contributed by NM, 22-Nov-1994.)

Theoremralnex 2715 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)

Theoremrexnal 2716 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)

Theoremdfral2 2717 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)

Theoremdfrex2 2718 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)

Theoremralbida 2719 Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)

Theoremrexbida 2720 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)

Theoremralbidva 2721* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 4-Mar-1997.)

Theoremrexbidva 2722* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 9-Mar-1997.)

Theoremralbid 2723 Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)

Theoremrexbid 2724 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)

Theoremralbidv 2725* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 20-Nov-1994.)

Theoremrexbidv 2726* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 20-Nov-1994.)

Theoremralbidv2 2727* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Apr-1997.)

Theoremrexbidv2 2728* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 22-May-1999.)

Theoremralbii 2729 Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario Carneiro, 17-Oct-2016.)

Theoremrexbii 2730 Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario Carneiro, 17-Oct-2016.)

Theorem2ralbii 2731 Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)

Theorem2rexbii 2732 Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 11-Nov-1995.)

Theoremralbii2 2733 Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005.)

Theoremrexbii2 2734 Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.)

Theoremraleqbii 2735 Equality deduction for restricted universal quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)

Theoremrexeqbii 2736 Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)

Theoremralbiia 2737 Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000.)

Theoremrexbiia 2738 Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 26-Oct-1999.)

Theorem2rexbiia 2739* Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)

Theoremr2alf 2740* Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremr2exf 2741* Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremr2al 2742* Double restricted universal quantification. (Contributed by NM, 19-Nov-1995.)

Theoremr2ex 2743* Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.)

Theorem2ralbida 2744* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 24-Feb-2004.)

Theorem2ralbidva 2745* Formula-building rule for restricted universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)

Theorem2rexbidva 2746* Formula-building rule for restricted existential quantifiers (deduction rule). (Contributed by NM, 15-Dec-2004.)

Theorem2ralbidv 2747* Formula-building rule for restricted universal quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon Jaroszewicz, 16-Mar-2007.)

Theorem2rexbidv 2748* Formula-building rule for restricted existential quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.)

Theoremrexralbidv 2749* Formula-building rule for restricted quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.)

Theoremralinexa 2750 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)

Theoremrexanali 2751 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)

Theoremnrexralim 2752 Negation of a complex predicate calculus formula. (Contributed by FL, 31-Jul-2009.)

Theoremrisset 2753* Two ways to say " belongs to ." (Contributed by NM, 22-Nov-1994.)

Theoremhbral 2754 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy, 13-Dec-2009.)

Theoremhbra1 2755 is not free in . (Contributed by NM, 18-Oct-1996.)

Theoremnfra1 2756 is not free in . (Contributed by NM, 18-Oct-1996.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfrald 2757 Deduction version of nfral 2759. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfrexd 2758 Deduction version of nfrex 2761. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremnfral 2759 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfra2 2760* Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 28972. Contributed by Alan Sare 31-Dec-2011. (Contributed by NM, 31-Dec-2011.)

Theoremnfrex 2761 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfre1 2762 is not free in . (Contributed by NM, 19-Mar-1997.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremr3al 2763* Triple restricted universal quantification. (Contributed by NM, 19-Nov-1995.)

Theoremalral 2764 Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)

Theoremrexex 2765 Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)

Theoremrsp 2766 Restricted specialization. (Contributed by NM, 17-Oct-1996.)

Theoremrspe 2767 Restricted specialization. (Contributed by NM, 12-Oct-1999.)

Theoremrsp2 2768 Restricted specialization. (Contributed by NM, 11-Feb-1997.)

Theoremrsp2e 2769 Restricted specialization. (Contributed by FL, 4-Jun-2012.)

Theoremrspec 2770 Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)

Theoremrgen 2771 Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)

Theoremrgen2a 2772* Generalization rule for restricted quantification. Note that and needn't be distinct (and illustrates the use of dvelim 2073). (Contributed by NM, 23-Nov-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.

Theoremrgenw 2773 Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)

Theoremrgen2w 2774 Generalization rule for restricted quantification. Note that and needn't be distinct. (Contributed by NM, 18-Jun-2014.)

Theoremmprg 2775 Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)

Theoremmprgbir 2776 Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.)

Theoremralim 2777 Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.)

Theoremralimi2 2778 Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)

Theoremralimia 2779 Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)

Theoremralimiaa 2780 Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.)

Theoremralimi 2781 Inference quantifying both antecedent and consequent, with strong hypothesis. (Contributed by NM, 4-Mar-1997.)

Theoremral2imi 2782 Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.)

Theoremralimdaa 2783 Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.)

Theoremralimdva 2784* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)

Theoremralimdv 2785* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 8-Oct-2003.)

Theoremralimdv2 2786* Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005.)

Theoremralrimi 2787 Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.)

Theoremralrimiv 2788* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.)

Theoremralrimiva 2789* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Jan-2006.)

Theoremralrimivw 2790* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)

Theoremr19.21t 2791 Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed theorem version). (Contributed by NM, 1-Mar-2008.)

Theoremr19.21 2792 Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 30-Mar-2011.)

Theoremr19.21v 2793* Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Theoremralrimd 2794 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.)

Theoremralrimdv 2795* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 27-May-1998.)

Theoremralrimdva 2796* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Feb-2008.)

Theoremralrimivv 2797* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 24-Jul-2004.)

Theoremralrimivva 2798* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremralrimivvva 2799* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with triple quantification.) (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremralrimdvv 2800* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 1-Jun-2005.)

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