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Theorem List for Metamath Proof Explorer - 2901-3000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrmobii 2901 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)

Theoremraleqf 2902 Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)

Theoremrexeqf 2903 Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)

Theoremreueq1f 2904 Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)

Theoremrmoeq1f 2905 Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremraleq 2906* Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)

Theoremrexeq 2907* Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)

Theoremreueq1 2908* Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)

Theoremrmoeq1 2909* Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremraleqi 2910* Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremrexeqi 2911* Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremraleqdv 2912* Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)

Theoremrexeqdv 2913* Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)

Theoremraleqbi1dv 2914* Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)

Theoremrexeqbi1dv 2915* Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)

Theoremreueqd 2916* Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)

Theoremrmoeqd 2917* Equality deduction for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremraleqbidv 2918* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)

Theoremrexeqbidv 2919* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)

Theoremraleqbidva 2920* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)

Theoremrexeqbidva 2921* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)

Theoremmormo 2922 Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)

Theoremreu5 2923 Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)

Theoremreurex 2924 Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)

Theoremreurmo 2925 Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)

Theoremrmo5 2926 Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)

Theoremnrexrmo 2927 Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)

Theoremcbvralf 2928 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)

Theoremcbvrexf 2929 Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.)

Theoremcbvral 2930* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.)

Theoremcbvrex 2931* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremcbvreu 2932* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)

Theoremcbvrmo 2933* Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.)

Theoremcbvralv 2934* Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.)

Theoremcbvrexv 2935* Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.)

Theoremcbvreuv 2936* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremcbvrmov 2937* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremcbvraldva2 2938* Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremcbvrexdva2 2939* Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremcbvraldva 2940* Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremcbvrexdva 2941* Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremcbvral2v 2942* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.)

Theoremcbvrex2v 2943* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)

Theoremcbvral3v 2944* Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.)

Theoremcbvralsv 2945* Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)

Theoremcbvrexsv 2946* Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.)

Theoremsbralie 2947* Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)

Theoremrabbiia 2948 Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.)

Theoremrabbidva 2949* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)

Theoremrabbidv 2950* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.)

Theoremrabeqf 2951 Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)

Theoremrabeq 2952* Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)

Theoremrabeqbidv 2953* Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)

Theoremrabeqbidva 2954* Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)

Theoremrabeq2i 2955 Inference rule from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)

Theoremcbvrab 2956 Rule to change the bound variable in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.)

Theoremcbvrabv 2957* Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.)

2.1.6  The universal class

Syntaxcvv 2958 Extend class notation to include the universal class symbol.

Theoremvjust 2959 Soundness justification theorem for df-v 2960. (Contributed by Rodolfo Medina, 27-Apr-2010.)

Definitiondf-v 2960 Define the universal class. Definition 5.20 of [TakeutiZaring] p. 21. Also Definition 2.9 of [Quine] p. 19. (Contributed by NM, 5-Aug-1993.)

Theoremvex 2961 All set variables are sets (see isset 2962). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.)

Theoremisset 2962* Two ways to say " is a set": A class is a member of the universal class (see df-v 2960) if and only if the class exists (i.e. there exists some set equal to class ). Theorem 6.9 of [Quine] p. 43. Notational convention: We will use the notational device " " to mean " is a set" very frequently, for example in uniex 4707. Note the when is not a set, it is called a proper class. In some theorems, such as uniexg 4708, in order to shorten certain proofs we use the more general antecedent instead of to mean " is a set."

Note that a constant is implicitly considered distinct from all variables. This is why is not included in the distinct variable list, even though df-clel 2434 requires that the expression substituted for not contain . (Also, the Metamath spec does not allow constants in the distinct variable list.) (Contributed by NM, 5-Aug-1993.)

Theoremissetf 2963 A version of isset that does not require x and A to be distinct. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremisseti 2964* A way to say " is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)

Theoremissetri 2965* A way to say " is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)

Theoremelex 2966 If a class is a member of another class, it is a set. Theorem 6.12 of [Quine] p. 44. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremelexi 2967 If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.)

Theoremelisset 2968* An element of a class exists. (Contributed by NM, 1-May-1995.)

Theoremelex22 2969* If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)

Theoremelex2 2970* If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)

Theoremralv 2971 A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)

Theoremrexv 2972 An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)

Theoremreuv 2973 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)

Theoremrmov 2974 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremrabab 2975 A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremralcom4 2976* Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremrexcom4 2977* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremrexcom4a 2978* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)

Theoremrexcom4b 2979* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)

Theoremceqsalt 2980* Closed theorem version of ceqsalg 2982. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremceqsralt 2981* Restricted quantifier version of ceqsalt 2980. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremceqsalg 2982* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremceqsal 2983* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)

Theoremceqsalv 2984* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)

Theoremceqsralv 2985* Restricted quantifier version of ceqsalv 2984. (Contributed by NM, 21-Jun-2013.)

Theoremgencl 2986* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)

Theorem2gencl 2987* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)

Theorem3gencl 2988* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)

Theoremcgsexg 2989* Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)

Theoremcgsex2g 2990* Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)

Theoremcgsex4g 2991* An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)

Theoremceqsex 2992* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremceqsexv 2993* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)

Theoremceqsex2 2994* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)

Theoremceqsex2v 2995* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)

Theoremceqsex3v 2996* Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)

Theoremceqsex4v 2997* Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)

Theoremceqsex6v 2998* Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.)

Theoremceqsex8v 2999* Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)

Theoremgencbvex 3000* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

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