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Theorem List for Intuitionistic Logic Explorer - 2501-2600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreueq1 2501* Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
 
Theoremrmoeq1 2502* Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 
Theoremraleqi 2503* Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)
   =>   
 
Theoremrexeqi 2504* Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
   =>   
 
Theoremraleqdv 2505* Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
   =>   
 
Theoremrexeqdv 2506* Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
   =>   
 
Theoremraleqbi1dv 2507* Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
   =>   
 
Theoremrexeqbi1dv 2508* Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
   =>   
 
Theoremreueqd 2509* Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
   =>   
 
Theoremrmoeqd 2510* Equality deduction for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
   =>   
 
Theoremraleqbidv 2511* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
   &       =>   
 
Theoremrexeqbidv 2512* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
   &       =>   
 
Theoremraleqbidva 2513* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
   &       =>   
 
Theoremrexeqbidva 2514* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
   &       =>   
 
Theoremmormo 2515 Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
 
Theoremreu5 2516 Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)
 
Theoremreurex 2517 Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
 
Theoremreurmo 2518 Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)
 
Theoremrmo5 2519 Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
 
Theoremnrexrmo 2520 Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
 
Theoremcbvralf 2521 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)
 F/_   &     F/_   &     F/   &     F/   &       =>   
 
Theoremcbvrexf 2522 Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)
 F/_   &     F/_   &     F/   &     F/   &       =>   
 
Theoremcbvral 2523* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.)

 F/   &     F/   &       =>   
 
Theoremcbvrex 2524* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

 F/   &     F/   &       =>   
 
Theoremcbvreu 2525* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)

 F/   &     F/   &       =>   
 
Theoremcbvrmo 2526* Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.)

 F/   &     F/   &       =>   
 
Theoremcbvralv 2527* Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.)
   =>   
 
Theoremcbvrexv 2528* Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.)
   =>   
 
Theoremcbvreuv 2529* 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 2530* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
   =>   
 
Theoremcbvraldva2 2531* 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 2532* 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 2533* 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 2534* 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 2535* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.)
   &       =>   
 
Theoremcbvrex2v 2536* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)
   &       =>   
 
Theoremcbvral3v 2537* Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.)
   &       &       =>     C  C
 
Theoremcbvralsv 2538* Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)
 
Theoremcbvrexsv 2539* Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.)
 
Theoremsbralie 2540* Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)
   =>   
 
Theoremrabbiia 2541 Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.)
   =>    
 {  |  }  {  |  }
 
Theoremrabbidva 2542* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)
   =>     {  |  }  {  |  }
 
Theoremrabbidv 2543* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.)
   =>     {  |  }  {  |  }
 
Theoremrabeqf 2544 Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)
 F/_   &     F/_   =>     {  |  }  {  |  }
 
Theoremrabeq 2545* Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)
 {  |  }  {  |  }
 
Theoremrabeqbidv 2546* Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)
   &       =>     {  |  }  {  |  }
 
Theoremrabeqbidva 2547* Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
   &       =>     {  |  }  {  |  }
 
Theoremrabeq2i 2548 Inference rule from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)
 {  |  }   =>   
 
Theoremcbvrab 2549 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.)
 F/_   &     F/_   &     F/   &     F/   &       =>     {  |  }  {  |  }
 
Theoremcbvrabv 2550* 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 2551 Extend class notation to include the universal class symbol.
 _V
 
Theoremvjust 2552 Soundness justification theorem for df-v 2553. (Contributed by Rodolfo Medina, 27-Apr-2010.)

 {  |  }  {  |  }
 
Definitiondf-v 2553 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.)

 _V  {  |  }
 
Theoremvex 2554 All setvar variables are sets (see isset 2555). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.)
 _V
 
Theoremisset 2555* Two ways to say " is a set": A class is a member of the universal class  _V (see df-v 2553) 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 "  _V " to mean " is a set" very frequently, for example in uniex . Note the when is not a set, it is called a proper class. In some theorems, such as uniexg , in order to shorten certain proofs we use the more general antecedent  V instead of  _V to mean " is a set."

Note that a constant is implicitly considered distinct from all variables. This is why  _V is not included in the distinct variable list, even though df-clel 2033 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.)

 _V
 
Theoremissetf 2556 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.)
 F/_   =>     _V
 
Theoremisseti 2557* A way to say " is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)
 _V   =>   
 
Theoremissetri 2558* A way to say " is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)
   =>     _V
 
Theoremeqvisset 2559 A class equal to a variable is a set. Note the absence of dv condition, contrary to isset 2555 and issetri 2558. (Contributed by BJ, 27-Apr-2019.)
 _V
 
Theoremelex 2560 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.)
 _V
 
Theoremelexi 2561 If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.)
   =>     _V
 
Theoremelisset 2562* An element of a class exists. (Contributed by NM, 1-May-1995.)
 V
 
Theoremelex22 2563* If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)
 C  C
 
Theoremelex2 2564* If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)
 
Theoremralv 2565 A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
 _V
 
Theoremrexv 2566 An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
 _V
 
Theoremreuv 2567 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
 _V
 
Theoremrmov 2568 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 _V
 
Theoremrabab 2569 A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

 {  _V  |  }  {  |  }
 
Theoremralcom4 2570* Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 
Theoremrexcom4 2571* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 
Theoremrexcom4a 2572* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
 
Theoremrexcom4b 2573* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
 _V   =>   
 
Theoremceqsalt 2574* Closed theorem version of ceqsalg 2576. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
 F/  V
 
Theoremceqsralt 2575* Restricted quantifier version of ceqsalt 2574. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
 F/
 
Theoremceqsalg 2576* 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.)

 F/   &       =>     V
 
Theoremceqsal 2577* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)

 F/   &     _V   &       =>   
 
Theoremceqsalv 2578* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
 _V   &       =>   
 
Theoremceqsralv 2579* Restricted quantifier version of ceqsalv 2578. (Contributed by NM, 21-Jun-2013.)
   =>   
 
Theoremgencl 2580* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
   &       &       =>   
 
Theorem2gencl 2581* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
 C  S  R  C   &     D  S  R  D   &     C    &     D    &     R  R    =>     C  S  D  S
 
Theorem3gencl 2582* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
 D  S  R  D   &     F  S  R  F   &     G  S  R  C  G   &     D    &     F    &     C  G    &     R  R  R    =>     D  S  F  S  G  S
 
Theoremcgsexg 2583* Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)
   &       =>     V
 
Theoremcgsex2g 2584* Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)
   &       =>     V  W
 
Theoremcgsex4g 2585* An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)
 C  D    &       =>     R  S  C  R  D  S
 
Theoremceqsex 2586* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)

 F/   &     _V   &       =>   
 
Theoremceqsexv 2587* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)
 _V   &       =>   
 
Theoremceqsex2 2588* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)

 F/   &     F/   &     _V   &     _V   &       &       =>   
 
Theoremceqsex2v 2589* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
 _V   &     _V   &       &       =>   
 
Theoremceqsex3v 2590* Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)
 _V   &     _V   &     C  _V   &       &       &     C    =>     C
 
Theoremceqsex4v 2591* Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
 _V   &     _V   &     C  _V   &     D  _V   &       &       &     C    &     D    =>     C  D
 
Theoremceqsex6v 2592* Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.)
 _V   &     _V   &     C  _V   &     D  _V   &     E  _V   &     F  _V   &       &       &     C    &     D    &     E    &     F    =>     C  D  E  F
 
Theoremceqsex8v 2593* Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
 _V   &     _V   &     C  _V   &     D  _V   &     E  _V   &     F  _V   &     G  _V   &     H  _V   &       &       &     C    &     D    &     E    &     F    &     t  G    &     s  H    =>     t s  C  D  E  F  t  G  s  H
 
Theoremgencbvex 2594* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 _V   &       &       &       =>   
 
Theoremgencbvex2 2595* Restatement of gencbvex 2594 with weaker hypotheses. (Contributed by Jeff Hankins, 6-Dec-2006.)
 _V   &       &       &       =>   
 
Theoremgencbval 2596* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof rewritten by Jim Kingdon, 20-Jun-2018.)
 _V   &       &       &       =>   
 
Theoremsbhypf 2597* Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (Contributed by Raph Levien, 10-Apr-2004.)

 F/   &       =>   
 
Theoremvtoclgft 2598 Closed theorem form of vtoclgf 2606. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)
 F/_  F/  V
 
Theoremvtocldf 2599 Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
 V   &       &       &     F/   &     F/_   &     F/   =>   
 
Theoremvtocld 2600* Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
 V   &       &       =>   
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