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

Theoremelexi 2701 If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.)

Theoremelexd 2702 If a class is a member of another class, it is a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Theoremelisset 2703* An element of a class exists. (Contributed by NM, 1-May-1995.)

Theoremelex22 2704* If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)

Theoremelex2 2705* If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)

Theoremralv 2706 A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)

Theoremrexv 2707 An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)

Theoremreuv 2708 A unique existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)

Theoremrmov 2709 An at-most-one quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremrabab 2710 A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremralcom4 2711* Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremrexcom4 2712* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremrexcom4a 2713* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)

Theoremrexcom4b 2714* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)

Theoremceqsalt 2715* Closed theorem version of ceqsalg 2717. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremceqsralt 2716* Restricted quantifier version of ceqsalt 2715. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremceqsalg 2717* 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 2718* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)

Theoremceqsalv 2719* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)

Theoremceqsralv 2720* Restricted quantifier version of ceqsalv 2719. (Contributed by NM, 21-Jun-2013.)

Theoremgencl 2721* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)

Theorem2gencl 2722* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)

Theorem3gencl 2723* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)

Theoremcgsexg 2724* Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)

Theoremcgsex2g 2725* Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)

Theoremcgsex4g 2726* An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)

Theoremceqsex 2727* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremceqsexv 2728* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)

Theoremceqsex2 2729* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)

Theoremceqsex2v 2730* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)

Theoremceqsex3v 2731* Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)

Theoremceqsex4v 2732* Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)

Theoremceqsex6v 2733* Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.)

Theoremceqsex8v 2734* Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)

Theoremgencbvex 2735* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremgencbvex2 2736* Restatement of gencbvex 2735 with weaker hypotheses. (Contributed by Jeff Hankins, 6-Dec-2006.)

Theoremgencbval 2737* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof rewritten by Jim Kingdon, 20-Jun-2018.)

Theoremsbhypf 2738* Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (Contributed by Raph Levien, 10-Apr-2004.)

Theoremvtoclgft 2739 Closed theorem form of vtoclgf 2747. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremvtocldf 2740 Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)

Theoremvtocld 2741* Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)

Theoremvtoclf 2742* Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1731. (Contributed by NM, 30-Aug-1993.)

Theoremvtocl 2743* Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993.)

Theoremvtocl2 2744* Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremvtocl3 2745* Implicit substitution of classes for setvar variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremvtoclb 2746* Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.)

Theoremvtoclgf 2747 Implicit substitution of a class for a setvar 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.)

Theoremvtoclg1f 2748* Version of vtoclgf 2747 with one non-freeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-11 1485 and ax-13 1492. (Contributed by BJ, 1-May-2019.)

Theoremvtoclg 2749* Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.)

Theoremvtoclbg 2750* Implicit substitution of a class for a setvar variable. (Contributed by NM, 29-Apr-1994.)

Theoremvtocl2gf 2751 Implicit substitution of a class for a setvar variable. (Contributed by NM, 25-Apr-1995.)

Theoremvtocl3gf 2752 Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremvtocl2g 2753* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.)

Theoremvtoclgaf 2754* Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremvtoclga 2755* Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.)

Theoremvtocl2gaf 2756* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.)

Theoremvtocl2ga 2757* Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.)

Theoremvtocl3gaf 2758* Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremvtocl3ga 2759* Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.)

Theoremvtocleg 2760* Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Jan-2004.)

Theoremvtoclegft 2761* Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2762.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremvtoclef 2762* Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.)

Theoremvtocle 2763* Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.)

Theoremvtoclri 2764* Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.)

Theoremspcimgft 2765 A closed version of spcimgf 2769. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcgft 2766 A closed version of spcgf 2771. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)

Theoremspcimegft 2767 A closed version of spcimegf 2770. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcegft 2768 A closed version of spcegf 2772. (Contributed by Jim Kingdon, 22-Jun-2018.)

Theoremspcimgf 2769 Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcimegf 2770 Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcgf 2771 Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 2-Feb-1997.) (Revised by Andrew Salmon, 12-Aug-2011.)

Theoremspcegf 2772 Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)

Theoremspcimdv 2773* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcdv 2774* Rule of specialization, using implicit substitution. Analogous to rspcdv 2795. (Contributed by David Moews, 1-May-2017.)

Theoremspcimedv 2775* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcgv 2776* Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)

Theoremspcegv 2777* Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)

Theoremspc2egv 2778* Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)

Theoremspc2gv 2779* Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)

Theoremspc3egv 2780* Existential specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)

Theoremspc3gv 2781* Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)

Theoremspcv 2782* Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)

Theoremspcev 2783* Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)

Theoremspc2ev 2784* Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)

Theoremrspct 2785* A closed version of rspc 2786. (Contributed by Andrew Salmon, 6-Jun-2011.)

Theoremrspc 2786* Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremrspce 2787* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremrspcv 2788* Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)

Theoremrspccv 2789* Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)

Theoremrspcva 2790* Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)

Theoremrspccva 2791* Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremrspcev 2792* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)

Theoremrspcimdv 2793* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremrspcimedv 2794* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremrspcdv 2795* Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)

Theoremrspcedv 2796* Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)

Theoremrspcdva 2797* Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 21-Jun-2020.)

Theoremrspcedvd 2798* Restricted existential specialization, using implicit substitution. Variant of rspcedv 2796. (Contributed by AV, 27-Nov-2019.)

Theoremrspcime 2799* Prove a restricted existential. (Contributed by Rohan Ridenour, 3-Aug-2023.)

Theoremrspceaimv 2800* Restricted existential specialization of a universally quantified implication. (Contributed by BJ, 24-Aug-2022.)

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