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

Theoremraleqbidva 2701* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)

Theoremrexeqbidva 2702* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)

Theoremcbvralf 2703 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)

Theoremcbvrexf 2704 Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.)

Theoremcbvral 2705* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.)

Theoremcbvrex 2706* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremcbvreu 2707* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)

Theoremcbvralv 2708* Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.)

Theoremcbvrexv 2709* Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.)

Theoremcbvreuv 2710* 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.)

Theoremcbvral2v 2711* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.)

Theoremcbvrex2v 2712* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)

Theoremcbvral3v 2713* Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.)

Theoremcbvralsv 2714* Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)

Theoremcbvrexsv 2715* Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.)

Theoremsbralie 2716* Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)

Theoremrabbiia 2717 Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.)

Theoremrabbidva 2718* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)

Theoremrabbidv 2719* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.)

Theoremrabeqf 2720 Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)

Theoremrabeq 2721* Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)

Theoremrabeqbidv 2722* Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)

Theoremrabeqbidva 2723* Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)

Theoremrabeq2i 2724 Inference rule from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)

Theoremcbvrab 2725 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 2726* 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 2727 Extend class notation to include the universal class symbol.

Theoremvjust 2728 Soundness justification theorem for df-v 2729. (Contributed by Rodolfo Medina, 27-Apr-2010.)

Definitiondf-v 2729 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 2730 All set variables are sets (see isset 2731). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.)

Theoremisset 2731* Two ways to say " is a set": A class is a member of the universal class (see df-v 2729) 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 4407. Note the when is not a set, it is called a proper class. In some theorems, such as uniexg 4408, 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 2249 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 2732 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 2733* A way to say " is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)

Theoremissetri 2734* A way to say " is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)

Theoremelex 2735 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 2736 If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.)

Theoremelisset 2737* An element of a class exists. (Contributed by NM, 1-May-1995.)

Theoremelex22 2738* If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)

Theoremelex2 2739* If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)

Theoremralv 2740 A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)

Theoremrexv 2741 An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)

Theoremreuv 2742 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)

Theoremrabab 2743 A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremralcom4 2744* Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremrexcom4 2745* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremrexcom4a 2746* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)

Theoremrexcom4b 2747* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)

Theoremceqsalt 2748* Closed theorem version of ceqsalg 2750. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremceqsralt 2749* Restricted quantifier version of ceqsalt 2748. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremceqsalg 2750* 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 2751* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)

Theoremceqsalv 2752* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)

Theoremceqsralv 2753* Restricted quantifier version of ceqsalv 2752. (Contributed by NM, 21-Jun-2013.)

Theoremgencl 2754* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)

Theorem2gencl 2755* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)

Theorem3gencl 2756* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)

Theoremcgsexg 2757* Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)

Theoremcgsex2g 2758* Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)

Theoremcgsex4g 2759* An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)

Theoremceqsex 2760* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremceqsexv 2761* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)

Theoremceqsex2 2762* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)

Theoremceqsex2v 2763* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)

Theoremceqsex3v 2764* Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)

Theoremceqsex4v 2765* Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)

Theoremceqsex6v 2766* Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.)

Theoremceqsex8v 2767* Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)

Theoremgencbvex 2768* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremgencbvex2 2769* Restatement of gencbvex 2768 with weaker hypotheses. (Contributed by Jeffrey Hankins, 6-Dec-2006.)

Theoremgencbval 2770* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.)

Theoremsbhypf 2771* Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3044. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremvtoclgft 2772 Closed theorem form of vtoclgf 2780. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremvtocldf 2773 Implicit substitution of a class for a set variable. (Contributed by Mario Carneiro, 15-Oct-2016.)

Theoremvtocld 2774* Implicit substitution of a class for a set variable. (Contributed by Mario Carneiro, 15-Oct-2016.)

Theoremvtoclf 2775* Implicit substitution of a class for a set variable. This is a generalization of chvar 1878. (Contributed by NM, 30-Aug-1993.)

Theoremvtocl 2776* Implicit substitution of a class for a set variable. (Contributed by NM, 30-Aug-1993.)

Theoremvtocl2 2777* Implicit substitution of classes for set variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremvtocl3 2778* Implicit substitution of classes for set variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremvtoclb 2779* Implicit substitution of a class for a set variable. (Contributed by NM, 23-Dec-1993.)

Theoremvtoclgf 2780 Implicit substitution of a class for a set variable, with bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.)

Theoremvtoclg 2781* Implicit substitution of a class expression for a set variable. (Contributed by NM, 17-Apr-1995.)

Theoremvtoclbg 2782* Implicit substitution of a class for a set variable. (Contributed by NM, 29-Apr-1994.)

Theoremvtocl2gf 2783 Implicit substitution of a class for a set variable. (Contributed by NM, 25-Apr-1995.)

Theoremvtocl3gf 2784 Implicit substitution of a class for a set variable. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremvtocl2g 2785* Implicit substitution of 2 classes for 2 set variables. (Contributed by NM, 25-Apr-1995.)

Theoremvtoclgaf 2786* Implicit substitution of a class for a set variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremvtoclga 2787* Implicit substitution of a class for a set variable. (Contributed by NM, 20-Aug-1995.)

Theoremvtocl2gaf 2788* Implicit substitution of 2 classes for 2 set variables. (Contributed by NM, 10-Aug-2013.)

Theoremvtocl2ga 2789* Implicit substitution of 2 classes for 2 set variables. (Contributed by NM, 20-Aug-1995.)

Theoremvtocl3gaf 2790* Implicit substitution of 3 classes for 3 set variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremvtocl3ga 2791* Implicit substitution of 3 classes for 3 set variables. (Contributed by NM, 20-Aug-1995.)

Theoremvtocleg 2792* Implicit substitution of a class for a set variable. (Contributed by NM, 10-Jan-2004.)

Theoremvtoclegft 2793* Implicit substitution of a class for a set variable. (Closed theorem version of vtoclef 2794.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremvtoclef 2794* Implicit substitution of a class for a set variable. (Contributed by NM, 18-Aug-1993.)

Theoremvtocle 2795* Implicit substitution of a class for a set variable. (Contributed by NM, 9-Sep-1993.)

Theoremvtoclri 2796* Implicit substitution of a class for a set variable. (Contributed by NM, 21-Nov-1994.)

Theoremcla4imgft 2797 A closed version of cla4imgf 2799. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremcla4gft 2798 A closed version of cla4gf 2801. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)

Theoremcla4imgf 2799 Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremcla4imegf 2800 Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

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