Home | Intuitionistic Logic Explorer Theorem List (p. 26 of 106) | < 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 | nfreuxy 2501* | Not-free for restricted uniqueness. This is a version where and are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.) |
Theorem | rabid 2502 | An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.) |
Theorem | rabid2 2503* | An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
Theorem | rabbi 2504 | Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2565. (Contributed by NM, 25-Nov-2013.) |
Theorem | rabswap 2505 | Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.) |
Theorem | nfrab1 2506 | The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.) |
Theorem | nfrabxy 2507* | A variable not free in a wff remains so in a restricted class abstraction. (Contributed by Jim Kingdon, 19-Jul-2018.) |
Theorem | reubida 2508 | Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by Mario Carneiro, 19-Nov-2016.) |
Theorem | reubidva 2509* | Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 13-Nov-2004.) |
Theorem | reubidv 2510* | Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 17-Oct-1996.) |
Theorem | reubiia 2511 | Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 14-Nov-2004.) |
Theorem | reubii 2512 | Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 22-Oct-1999.) |
Theorem | rmobida 2513 | Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.) |
Theorem | rmobidva 2514* | Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.) |
Theorem | rmobidv 2515* | Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.) |
Theorem | rmobiia 2516 | Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.) |
Theorem | rmobii 2517 | Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.) |
Theorem | raleqf 2518 | 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.) |
Theorem | rexeqf 2519 | 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.) |
Theorem | reueq1f 2520 | 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.) |
Theorem | rmoeq1f 2521 | Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | raleq 2522* | Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) |
Theorem | rexeq 2523* | Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.) |
Theorem | reueq1 2524* | Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.) |
Theorem | rmoeq1 2525* | Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | raleqi 2526* | Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | rexeqi 2527* | Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Theorem | raleqdv 2528* | Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.) |
Theorem | rexeqdv 2529* | Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.) |
Theorem | raleqbi1dv 2530* | Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) |
Theorem | rexeqbi1dv 2531* | Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.) |
Theorem | reueqd 2532* | Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.) |
Theorem | rmoeqd 2533* | Equality deduction for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | raleqbidv 2534* | Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) |
Theorem | rexeqbidv 2535* | Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) |
Theorem | raleqbidva 2536* | Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.) |
Theorem | rexeqbidva 2537* | Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.) |
Theorem | mormo 2538 | Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.) |
Theorem | reu5 2539 | Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.) |
Theorem | reurex 2540 | Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.) |
Theorem | reurmo 2541 | Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.) |
Theorem | rmo5 2542 | Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.) |
Theorem | nrexrmo 2543 | Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.) |
Theorem | cbvralf 2544 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.) |
Theorem | cbvrexf 2545 | 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 2546* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) |
Theorem | cbvrex 2547* | 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 2548* | Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Theorem | cbvrmo 2549* | Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.) |
Theorem | cbvralv 2550* | Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.) |
Theorem | cbvrexv 2551* | Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.) |
Theorem | cbvreuv 2552* | 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 2553* | Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | cbvraldva2 2554* | 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 2555* | 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 2556* | 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 2557* | 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 2558* | Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.) |
Theorem | cbvrex2v 2559* | Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.) |
Theorem | cbvral3v 2560* | Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.) |
Theorem | cbvralsv 2561* | Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.) |
Theorem | cbvrexsv 2562* | Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.) |
Theorem | sbralie 2563* | Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.) |
Theorem | rabbiia 2564 | Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.) |
Theorem | rabbidva 2565* | Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.) |
Theorem | rabbidv 2566* | Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.) |
Theorem | rabeqf 2567 | Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) |
Theorem | rabeq 2568* | Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.) |
Theorem | rabeqbidv 2569* | Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.) |
Theorem | rabeqbidva 2570* | Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.) |
Theorem | rabeq2i 2571 | Inference rule from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.) |
Theorem | cbvrab 2572 | 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 2573* | Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.) |
Syntax | cvv 2574 | Extend class notation to include the universal class symbol. |
Theorem | vjust 2575 | Soundness justification theorem for df-v 2576. (Contributed by Rodolfo Medina, 27-Apr-2010.) |
Definition | df-v 2576 | 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 2577 | All setvar variables are sets (see isset 2578). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.) |
Theorem | isset 2578* |
Two ways to say "
is a set": A class is a member of the
universal class (see df-v 2576) 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 4202. Note the when is not a set,
it is called a proper class. In some theorems, such as uniexg 4203, 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 2052 requires that the expression substituted for not contain . (Also, the Metamath spec does not allow constants in the distinct variable list.) (Contributed by NM, 26-May-1993.) |
Theorem | issetf 2579 | 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 2580* | A way to say " is a set" (inference rule). (Contributed by NM, 5-Aug-1993.) |
Theorem | issetri 2581* | A way to say " is a set" (inference rule). (Contributed by NM, 5-Aug-1993.) |
Theorem | eqvisset 2582 | A class equal to a variable is a set. Note the absence of dv condition, contrary to isset 2578 and issetri 2581. (Contributed by BJ, 27-Apr-2019.) |
Theorem | elex 2583 | 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 2584 | If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.) |
Theorem | elisset 2585* | An element of a class exists. (Contributed by NM, 1-May-1995.) |
Theorem | elex22 2586* | If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.) |
Theorem | elex2 2587* | If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.) |
Theorem | ralv 2588 | A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.) |
Theorem | rexv 2589 | An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.) |
Theorem | reuv 2590 | A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.) |
Theorem | rmov 2591 | A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | rabab 2592 | 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 2593* | Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | rexcom4 2594* | Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | rexcom4a 2595* | Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.) |
Theorem | rexcom4b 2596* | Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.) |
Theorem | ceqsalt 2597* | Closed theorem version of ceqsalg 2599. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) |
Theorem | ceqsralt 2598* | Restricted quantifier version of ceqsalt 2597. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) |
Theorem | ceqsalg 2599* | 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 2600* | A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |