Home | Intuitionistic Logic Explorer Theorem List (p. 26 of 95) | < Previous Next > |
Browser slow? Try the
Unicode version. |
||
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | reueq1 2501* | Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.) |
Theorem | rmoeq1 2502* | Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | raleqi 2503* | Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | rexeqi 2504* | Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Theorem | raleqdv 2505* | Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.) |
Theorem | rexeqdv 2506* | Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.) |
Theorem | raleqbi1dv 2507* | Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) |
Theorem | rexeqbi1dv 2508* | Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.) |
Theorem | reueqd 2509* | Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.) |
Theorem | rmoeqd 2510* | Equality deduction for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | raleqbidv 2511* | Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) |
Theorem | rexeqbidv 2512* | Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) |
Theorem | raleqbidva 2513* | Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.) |
Theorem | rexeqbidva 2514* | Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.) |
Theorem | mormo 2515 | Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.) |
Theorem | reu5 2516 | Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.) |
Theorem | reurex 2517 | Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.) |
Theorem | reurmo 2518 | Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.) |
Theorem | rmo5 2519 | Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.) |
Theorem | nrexrmo 2520 | Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.) |
Theorem | cbvralf 2521 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.) |
Theorem | cbvrexf 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.) |
Theorem | cbvral 2523* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) |
Theorem | cbvrex 2524* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | cbvreu 2525* | Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Theorem | cbvrmo 2526* | Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.) |
Theorem | cbvralv 2527* | Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.) |
Theorem | cbvrexv 2528* | Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.) |
Theorem | cbvreuv 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.) |
Theorem | cbvrmov 2530* | Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | cbvraldva2 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.) |
Theorem | cbvrexdva2 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.) |
Theorem | cbvraldva 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.) |
Theorem | cbvrexdva 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.) |
Theorem | cbvral2v 2535* | Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.) |
Theorem | cbvrex2v 2536* | Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.) |
Theorem | cbvral3v 2537* | Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.) |
Theorem | cbvralsv 2538* | Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.) |
Theorem | cbvrexsv 2539* | Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.) |
Theorem | sbralie 2540* | Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.) |
Theorem | rabbiia 2541 | Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.) |
Theorem | rabbidva 2542* | Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.) |
Theorem | rabbidv 2543* | Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.) |
Theorem | rabeqf 2544 | Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) |
Theorem | rabeq 2545* | Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.) |
Theorem | rabeqbidv 2546* | Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.) |
Theorem | rabeqbidva 2547* | Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.) |
Theorem | rabeq2i 2548 | Inference rule from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.) |
Theorem | cbvrab 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.) |
Theorem | cbvrabv 2550* | Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.) |
Syntax | cvv 2551 | Extend class notation to include the universal class symbol. |
Theorem | vjust 2552 | Soundness justification theorem for df-v 2553. (Contributed by Rodolfo Medina, 27-Apr-2010.) |
Definition | df-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.) |
Theorem | vex 2554 | All setvar variables are sets (see isset 2555). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.) |
Theorem | isset 2555* |
Two ways to say " is a set": A class is a member of the
universal class (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 " " 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
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 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.) |
Theorem | issetf 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.) |
Theorem | isseti 2557* | A way to say " is a set" (inference rule). (Contributed by NM, 5-Aug-1993.) |
Theorem | issetri 2558* | A way to say " is a set" (inference rule). (Contributed by NM, 5-Aug-1993.) |
Theorem | eqvisset 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.) |
Theorem | elex 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.) |
Theorem | elexi 2561 | If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.) |
Theorem | elisset 2562* | An element of a class exists. (Contributed by NM, 1-May-1995.) |
Theorem | elex22 2563* | If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.) |
Theorem | elex2 2564* | If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.) |
Theorem | ralv 2565 | A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.) |
Theorem | rexv 2566 | An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.) |
Theorem | reuv 2567 | A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.) |
Theorem | rmov 2568 | A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | rabab 2569 | A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | ralcom4 2570* | Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | rexcom4 2571* | Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | rexcom4a 2572* | Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.) |
Theorem | rexcom4b 2573* | Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.) |
Theorem | ceqsalt 2574* | Closed theorem version of ceqsalg 2576. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) |
Theorem | ceqsralt 2575* | Restricted quantifier version of ceqsalt 2574. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) |
Theorem | ceqsalg 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.) |
Theorem | ceqsal 2577* | A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.) |
Theorem | ceqsalv 2578* | A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.) |
Theorem | ceqsralv 2579* | Restricted quantifier version of ceqsalv 2578. (Contributed by NM, 21-Jun-2013.) |
Theorem | gencl 2580* | Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.) |
Theorem | 2gencl 2581* | Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.) |
Theorem | 3gencl 2582* | Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.) |
Theorem | cgsexg 2583* | Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.) |
Theorem | cgsex2g 2584* | Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.) |
Theorem | cgsex4g 2585* | An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.) |
Theorem | ceqsex 2586* | Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.) |
Theorem | ceqsexv 2587* | Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) |
Theorem | ceqsex2 2588* | Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.) |
Theorem | ceqsex2v 2589* | Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.) |
Theorem | ceqsex3v 2590* | Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.) |
Theorem | ceqsex4v 2591* | Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.) |
Theorem | ceqsex6v 2592* | Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.) |
Theorem | ceqsex8v 2593* | Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.) |
Theorem | gencbvex 2594* | Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | gencbvex2 2595* | Restatement of gencbvex 2594 with weaker hypotheses. (Contributed by Jeff Hankins, 6-Dec-2006.) |
Theorem | gencbval 2596* | Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof rewritten by Jim Kingdon, 20-Jun-2018.) |
Theorem | sbhypf 2597* | Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (Contributed by Raph Levien, 10-Apr-2004.) |
Theorem | vtoclgft 2598 | Closed theorem form of vtoclgf 2606. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | vtocldf 2599 | Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Theorem | vtocld 2600* | Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |