HomeHome Metamath Proof Explorer
Theorem List (p. 439 of 482)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-30715)
  Hilbert Space Explorer  Hilbert Space Explorer
(30716-32238)
  Users' Mathboxes  Users' Mathboxes
(32239-48161)
 

Theorem List for Metamath Proof Explorer - 43801-43900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcompne 43801 The complement of ๐ด is not equal to ๐ด. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by BJ, 11-Nov-2021.)
(V โˆ– ๐ด) โ‰  ๐ด
 
Theoremcompab 43802 Two ways of saying "the complement of a class abstraction". (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
(V โˆ– {๐‘ง โˆฃ ๐œ‘}) = {๐‘ง โˆฃ ยฌ ๐œ‘}
 
Theoremconss2 43803 Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
(๐ด โІ (V โˆ– ๐ต) โ†” ๐ต โІ (V โˆ– ๐ด))
 
Theoremconss1 43804 Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
((V โˆ– ๐ด) โІ ๐ต โ†” (V โˆ– ๐ต) โІ ๐ด)
 
Theoremralbidar 43805 More general form of ralbida 3262. (Contributed by Andrew Salmon, 25-Jul-2011.)
(๐œ‘ โ†’ โˆ€๐‘ฅ โˆˆ ๐ด ๐œ‘)    &   ((๐œ‘ โˆง ๐‘ฅ โˆˆ ๐ด) โ†’ (๐œ“ โ†” ๐œ’))    โ‡’   (๐œ‘ โ†’ (โˆ€๐‘ฅ โˆˆ ๐ด ๐œ“ โ†” โˆ€๐‘ฅ โˆˆ ๐ด ๐œ’))
 
Theoremrexbidar 43806 More general form of rexbida 3264. (Contributed by Andrew Salmon, 25-Jul-2011.)
(๐œ‘ โ†’ โˆ€๐‘ฅ โˆˆ ๐ด ๐œ‘)    &   ((๐œ‘ โˆง ๐‘ฅ โˆˆ ๐ด) โ†’ (๐œ“ โ†” ๐œ’))    โ‡’   (๐œ‘ โ†’ (โˆƒ๐‘ฅ โˆˆ ๐ด ๐œ“ โ†” โˆƒ๐‘ฅ โˆˆ ๐ด ๐œ’))
 
Theoremdropab1 43807 Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)
(โˆ€๐‘ฅ ๐‘ฅ = ๐‘ฆ โ†’ {โŸจ๐‘ฅ, ๐‘งโŸฉ โˆฃ ๐œ‘} = {โŸจ๐‘ฆ, ๐‘งโŸฉ โˆฃ ๐œ‘})
 
Theoremdropab2 43808 Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)
(โˆ€๐‘ฅ ๐‘ฅ = ๐‘ฆ โ†’ {โŸจ๐‘ง, ๐‘ฅโŸฉ โˆฃ ๐œ‘} = {โŸจ๐‘ง, ๐‘ฆโŸฉ โˆฃ ๐œ‘})
 
Theoremipo0 43809 If the identity relation partially orders any class, then that class is the null class. (Contributed by Andrew Salmon, 25-Jul-2011.)
( I Po ๐ด โ†” ๐ด = โˆ…)
 
Theoremifr0 43810 A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011.)
( I Fr ๐ด โ†” ๐ด = โˆ…)
 
Theoremordpss 43811 ordelpss 6391 with an antecedent removed. (Contributed by Andrew Salmon, 25-Jul-2011.)
(Ord ๐ต โ†’ (๐ด โˆˆ ๐ต โ†’ ๐ด โŠŠ ๐ต))
 
Theoremfvsb 43812* Explicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)
(โˆƒ!๐‘ฆ ๐ด๐น๐‘ฆ โ†’ ([(๐นโ€˜๐ด) / ๐‘ฅ]๐œ‘ โ†” โˆƒ๐‘ฅ(โˆ€๐‘ฆ(๐ด๐น๐‘ฆ โ†” ๐‘ฆ = ๐‘ฅ) โˆง ๐œ‘)))
 
Theoremfveqsb 43813* Implicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)
(๐‘ฅ = (๐นโ€˜๐ด) โ†’ (๐œ‘ โ†” ๐œ“))    &   โ„ฒ๐‘ฅ๐œ“    โ‡’   (โˆƒ!๐‘ฆ ๐ด๐น๐‘ฆ โ†’ (๐œ“ โ†” โˆƒ๐‘ฅ(โˆ€๐‘ฆ(๐ด๐น๐‘ฆ โ†” ๐‘ฆ = ๐‘ฅ) โˆง ๐œ‘)))
 
Theoremxpexb 43814 A Cartesian product exists iff its converse does. Corollary 6.9(1) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)
((๐ด ร— ๐ต) โˆˆ V โ†” (๐ต ร— ๐ด) โˆˆ V)
 
Theoremtrelpss 43815 An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 5656, ax-reg 9607 is required for its proof. (Contributed by Andrew Salmon, 13-Nov-2011.)
((Tr ๐ด โˆง ๐ต โˆˆ ๐ด) โ†’ ๐ต โŠŠ ๐ด)
 
21.38.6  Arithmetic
 
Theoremaddcomgi 43816 Generalization of commutative law for addition. Simplifies proofs dealing with vectors. However, it is dependent on our particular definition of ordered pair. (Contributed by Andrew Salmon, 28-Jan-2012.) (Revised by Mario Carneiro, 6-May-2015.)
(๐ด + ๐ต) = (๐ต + ๐ด)
 
21.38.7  Geometry
 
Syntaxcplusr 43817 Introduce the operation of vector addition.
class +๐‘Ÿ
 
Syntaxcminusr 43818 Introduce the operation of vector subtraction.
class -๐‘Ÿ
 
Syntaxctimesr 43819 Introduce the operation of scalar multiplication.
class .๐‘ฃ
 
Syntaxcptdfc 43820 PtDf is a predicate that is crucial for the definition of lines as well as proving a number of important theorems.
class PtDf(๐ด, ๐ต)
 
Syntaxcrr3c 43821 RR3 is a class.
class RR3
 
Syntaxcline3 43822 line3 is a class.
class line3
 
Definitiondf-addr 43823* Define the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.)
+๐‘Ÿ = (๐‘ฅ โˆˆ V, ๐‘ฆ โˆˆ V โ†ฆ (๐‘ฃ โˆˆ โ„ โ†ฆ ((๐‘ฅโ€˜๐‘ฃ) + (๐‘ฆโ€˜๐‘ฃ))))
 
Definitiondf-subr 43824* Define the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.)
-๐‘Ÿ = (๐‘ฅ โˆˆ V, ๐‘ฆ โˆˆ V โ†ฆ (๐‘ฃ โˆˆ โ„ โ†ฆ ((๐‘ฅโ€˜๐‘ฃ) โˆ’ (๐‘ฆโ€˜๐‘ฃ))))
 
Definitiondf-mulv 43825* Define the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.)
.๐‘ฃ = (๐‘ฅ โˆˆ V, ๐‘ฆ โˆˆ V โ†ฆ (๐‘ฃ โˆˆ โ„ โ†ฆ (๐‘ฅ ยท (๐‘ฆโ€˜๐‘ฃ))))
 
Theoremaddrval 43826* Value of the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.)
((๐ด โˆˆ ๐ถ โˆง ๐ต โˆˆ ๐ท) โ†’ (๐ด+๐‘Ÿ๐ต) = (๐‘ฃ โˆˆ โ„ โ†ฆ ((๐ดโ€˜๐‘ฃ) + (๐ตโ€˜๐‘ฃ))))
 
Theoremsubrval 43827* Value of the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.)
((๐ด โˆˆ ๐ถ โˆง ๐ต โˆˆ ๐ท) โ†’ (๐ด-๐‘Ÿ๐ต) = (๐‘ฃ โˆˆ โ„ โ†ฆ ((๐ดโ€˜๐‘ฃ) โˆ’ (๐ตโ€˜๐‘ฃ))))
 
Theoremmulvval 43828* Value of the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.)
((๐ด โˆˆ ๐ถ โˆง ๐ต โˆˆ ๐ท) โ†’ (๐ด.๐‘ฃ๐ต) = (๐‘ฃ โˆˆ โ„ โ†ฆ (๐ด ยท (๐ตโ€˜๐‘ฃ))))
 
Theoremaddrfv 43829 Vector addition at a value. The operation takes each vector ๐ด and ๐ต and forms a new vector whose values are the sum of each of the values of ๐ด and ๐ต. (Contributed by Andrew Salmon, 27-Jan-2012.)
((๐ด โˆˆ ๐ธ โˆง ๐ต โˆˆ ๐ท โˆง ๐ถ โˆˆ โ„) โ†’ ((๐ด+๐‘Ÿ๐ต)โ€˜๐ถ) = ((๐ดโ€˜๐ถ) + (๐ตโ€˜๐ถ)))
 
Theoremsubrfv 43830 Vector subtraction at a value. (Contributed by Andrew Salmon, 27-Jan-2012.)
((๐ด โˆˆ ๐ธ โˆง ๐ต โˆˆ ๐ท โˆง ๐ถ โˆˆ โ„) โ†’ ((๐ด-๐‘Ÿ๐ต)โ€˜๐ถ) = ((๐ดโ€˜๐ถ) โˆ’ (๐ตโ€˜๐ถ)))
 
Theoremmulvfv 43831 Scalar multiplication at a value. (Contributed by Andrew Salmon, 27-Jan-2012.)
((๐ด โˆˆ ๐ธ โˆง ๐ต โˆˆ ๐ท โˆง ๐ถ โˆˆ โ„) โ†’ ((๐ด.๐‘ฃ๐ต)โ€˜๐ถ) = (๐ด ยท (๐ตโ€˜๐ถ)))
 
Theoremaddrfn 43832 Vector addition produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
((๐ด โˆˆ ๐ถ โˆง ๐ต โˆˆ ๐ท) โ†’ (๐ด+๐‘Ÿ๐ต) Fn โ„)
 
Theoremsubrfn 43833 Vector subtraction produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
((๐ด โˆˆ ๐ถ โˆง ๐ต โˆˆ ๐ท) โ†’ (๐ด-๐‘Ÿ๐ต) Fn โ„)
 
Theoremmulvfn 43834 Scalar multiplication producees a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
((๐ด โˆˆ ๐ถ โˆง ๐ต โˆˆ ๐ท) โ†’ (๐ด.๐‘ฃ๐ต) Fn โ„)
 
Theoremaddrcom 43835 Vector addition is commutative. (Contributed by Andrew Salmon, 28-Jan-2012.)
((๐ด โˆˆ ๐ถ โˆง ๐ต โˆˆ ๐ท) โ†’ (๐ด+๐‘Ÿ๐ต) = (๐ต+๐‘Ÿ๐ด))
 
Definitiondf-ptdf 43836* Define the predicate PtDf, which is a utility definition used to shorten definitions and simplify proofs. (Contributed by Andrew Salmon, 15-Jul-2012.)
PtDf(๐ด, ๐ต) = (๐‘ฅ โˆˆ โ„ โ†ฆ (((๐‘ฅ.๐‘ฃ(๐ต-๐‘Ÿ๐ด)) +๐‘ฃ ๐ด) โ€œ {1, 2, 3}))
 
Definitiondf-rr3 43837 Define the set of all points RR3. We define each point ๐ด as a function to allow the use of vector addition and subtraction as well as scalar multiplication in our proofs. (Contributed by Andrew Salmon, 15-Jul-2012.)
RR3 = (โ„ โ†‘m {1, 2, 3})
 
Definitiondf-line3 43838* Define the set of all lines. A line is an infinite subset of RR3 that satisfies a PtDf property. (Contributed by Andrew Salmon, 15-Jul-2012.)
line3 = {๐‘ฅ โˆˆ ๐’ซ RR3 โˆฃ (2o โ‰ผ ๐‘ฅ โˆง โˆ€๐‘ฆ โˆˆ ๐‘ฅ โˆ€๐‘ง โˆˆ ๐‘ฅ (๐‘ง โ‰  ๐‘ฆ โ†’ ran PtDf(๐‘ฆ, ๐‘ง) = ๐‘ฅ))}
 
21.39  Mathbox for Alan Sare

We are sad to report the passing of long-time contributor Alan Sare (Nov. 9, 1954 - Mar. 23, 2019).

Alan's first contribution to Metamath was a shorter proof for tfrlem8 8398 in 2008.

He developed a tool called "completeusersproof" that assists developing proofs using his "virtual deduction" method: https://us.metamath.org/other.html#completeusersproof 8398. His virtual deduction method is explained in the comment for wvd1 43931.

Below are some excerpts from his first emails to NM in 2007:

...I have been interested in proving set theory theorems for many years for mental exercise. I enjoy it. I have used a book by Martin Zuckerman. It is informal. I am interested in completely and perfectly proving theorems. Mr. Zuckerman leaves out most of the steps of a proof, of course, like most authors do, as you have noted. A complete proof for higher theorems would require a volume of writing similar to the Metamath documents. So I am frustrated when I am not capable of constructing a proof and Zuckerman leaves out steps I do not understand. I could search for the steps in other texts, but I don't do that too much. Metamath may be the answer for me....

...If we go beyond mathematics, I believe that it is possible to write down all human knowledge in a way similar to the way you have explicated large areas of mathematics. Of course, that would be a much, much more difficult job. For example, it is possible to take a hard science like physics, construct axioms based on experimental results, and to cast all of physics into a collection of axioms and theorems. Maybe this has already been attempted, although I am not familiar with it. When one then moves on to the soft sciences such as social science, this job gets much more difficult. The key is: All human thought consists of logical operations on abstract objects. Usually, these logical operations are done informally. There is no reason why one cannot take any subject and explicate it and take it down to the indivisible postulates in a formal rigorous way....

...When I read a math book or an engineering book I come across something I don't understand and I am compelled to understand it. But, often it is hopeless. I don't have the time. Or, I would have to read the same thing by multiple authors in the hope that different authors would give parts of the working proof that others have omitted. It is very inefficient. Because I have always been inclined to "get to the bottom" for a 100% fully understood proof....

 
21.39.1  Auxiliary theorems for the Virtual Deduction tool
 
TheoremidiALT 43839 Placeholder for idi 1. Though unnecessary, this theorem is sometimes used in proofs in this mathbox for pedagogical purposes. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
๐œ‘    โ‡’   ๐œ‘
 
Theoremexbir 43840 Exportation implication also converting the consequent from a biconditional to an implication. Derived automatically from exbirVD 44215. (Contributed by Alan Sare, 31-Dec-2011.)
(((๐œ‘ โˆง ๐œ“) โ†’ (๐œ’ โ†” ๐œƒ)) โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ (๐œƒ โ†’ ๐œ’))))
 
Theorem3impexpbicom 43841 Version of 3impexp 1356 where in addition the consequent is commuted. (Contributed by Alan Sare, 31-Dec-2011.)
(((๐œ‘ โˆง ๐œ“ โˆง ๐œ’) โ†’ (๐œƒ โ†” ๐œ)) โ†” (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ (๐œ โ†” ๐œƒ)))))
 
Theorem3impexpbicomi 43842 Inference associated with 3impexpbicom 43841. Derived automatically from 3impexpbicomiVD 44220. (Contributed by Alan Sare, 31-Dec-2011.)
((๐œ‘ โˆง ๐œ“ โˆง ๐œ’) โ†’ (๐œƒ โ†” ๐œ))    โ‡’   (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ (๐œ โ†” ๐œƒ))))
 
21.39.2  Supplementary unification deductions
 
Theorembi1imp 43843 Importation inference similar to imp 406, except the outermost implication of the hypothesis is a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(๐œ‘ โ†” (๐œ“ โ†’ ๐œ’))    โ‡’   ((๐œ‘ โˆง ๐œ“) โ†’ ๐œ’)
 
Theorembi2imp 43844 Importation inference similar to imp 406, except both implications of the hypothesis are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(๐œ‘ โ†” (๐œ“ โ†” ๐œ’))    โ‡’   ((๐œ‘ โˆง ๐œ“) โ†’ ๐œ’)
 
Theorembi3impb 43845 Similar to 3impb 1113 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
((๐œ‘ โˆง (๐œ“ โˆง ๐œ’)) โ†” ๐œƒ)    โ‡’   ((๐œ‘ โˆง ๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)
 
Theorembi3impa 43846 Similar to 3impa 1108 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(((๐œ‘ โˆง ๐œ“) โˆง ๐œ’) โ†” ๐œƒ)    โ‡’   ((๐œ‘ โˆง ๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)
 
Theorembi23impib 43847 3impib 1114 with the inner implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(๐œ‘ โ†’ ((๐œ“ โˆง ๐œ’) โ†” ๐œƒ))    โ‡’   ((๐œ‘ โˆง ๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)
 
Theorembi13impib 43848 3impib 1114 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(๐œ‘ โ†” ((๐œ“ โˆง ๐œ’) โ†’ ๐œƒ))    โ‡’   ((๐œ‘ โˆง ๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)
 
Theorembi123impib 43849 3impib 1114 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(๐œ‘ โ†” ((๐œ“ โˆง ๐œ’) โ†” ๐œƒ))    โ‡’   ((๐œ‘ โˆง ๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)
 
Theorembi13impia 43850 3impia 1115 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
((๐œ‘ โˆง ๐œ“) โ†” (๐œ’ โ†’ ๐œƒ))    โ‡’   ((๐œ‘ โˆง ๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)
 
Theorembi123impia 43851 3impia 1115 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
((๐œ‘ โˆง ๐œ“) โ†” (๐œ’ โ†” ๐œƒ))    โ‡’   ((๐œ‘ โˆง ๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)
 
Theorembi33imp12 43852 3imp 1109 with innermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†” ๐œƒ)))    โ‡’   ((๐œ‘ โˆง ๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)
 
Theorembi23imp13 43853 3imp 1109 with middle implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(๐œ‘ โ†’ (๐œ“ โ†” (๐œ’ โ†’ ๐œƒ)))    โ‡’   ((๐œ‘ โˆง ๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)
 
Theorembi13imp23 43854 3imp 1109 with outermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(๐œ‘ โ†” (๐œ“ โ†’ (๐œ’ โ†’ ๐œƒ)))    โ‡’   ((๐œ‘ โˆง ๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)
 
Theorembi13imp2 43855 Similar to 3imp 1109 except the outermost and innermost implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(๐œ‘ โ†” (๐œ“ โ†’ (๐œ’ โ†” ๐œƒ)))    โ‡’   ((๐œ‘ โˆง ๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)
 
Theorembi12imp3 43856 Similar to 3imp 1109 except all but innermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(๐œ‘ โ†” (๐œ“ โ†” (๐œ’ โ†’ ๐œƒ)))    โ‡’   ((๐œ‘ โˆง ๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)
 
Theorembi23imp1 43857 Similar to 3imp 1109 except all but outermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(๐œ‘ โ†’ (๐œ“ โ†” (๐œ’ โ†” ๐œƒ)))    โ‡’   ((๐œ‘ โˆง ๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)
 
Theorembi123imp0 43858 Similar to 3imp 1109 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(๐œ‘ โ†” (๐œ“ โ†” (๐œ’ โ†” ๐œƒ)))    โ‡’   ((๐œ‘ โˆง ๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)
 
Theorem4animp1 43859 A single hypothesis unification deduction with an assertion which is an implication with a 4-right-nested conjunction antecedent. (Contributed by Alan Sare, 30-May-2018.)
((๐œ‘ โˆง ๐œ“ โˆง ๐œ’) โ†’ (๐œ โ†” ๐œƒ))    โ‡’   ((((๐œ‘ โˆง ๐œ“) โˆง ๐œ’) โˆง ๐œƒ) โ†’ ๐œ)
 
Theorem4an31 43860 A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.)
((((๐œ’ โˆง ๐œ“) โˆง ๐œ‘) โˆง ๐œƒ) โ†’ ๐œ)    โ‡’   ((((๐œ‘ โˆง ๐œ“) โˆง ๐œ’) โˆง ๐œƒ) โ†’ ๐œ)
 
Theorem4an4132 43861 A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.)
((((๐œƒ โˆง ๐œ’) โˆง ๐œ“) โˆง ๐œ‘) โ†’ ๐œ)    โ‡’   ((((๐œ‘ โˆง ๐œ“) โˆง ๐œ’) โˆง ๐œƒ) โ†’ ๐œ)
 
Theoremexpcomdg 43862 Biconditional form of expcomd 416. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.)
((๐œ‘ โ†’ ((๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)) โ†” (๐œ‘ โ†’ (๐œ’ โ†’ (๐œ“ โ†’ ๐œƒ))))
 
21.39.3  Conventional Metamath proofs, some derived from VD proofs
 
Theoremiidn3 43863 idn3 43977 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œ’)))
 
Theoremee222 43864 e222 43998 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 7-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(๐œ‘ โ†’ (๐œ“ โ†’ ๐œ’))    &   (๐œ‘ โ†’ (๐œ“ โ†’ ๐œƒ))    &   (๐œ‘ โ†’ (๐œ“ โ†’ ๐œ))    &   (๐œ’ โ†’ (๐œƒ โ†’ (๐œ โ†’ ๐œ‚)))    โ‡’   (๐œ‘ โ†’ (๐œ“ โ†’ ๐œ‚))
 
Theoremee3bir 43865 Right-biconditional form of e3 44099 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œƒ)))    &   (๐œ โ†” ๐œƒ)    โ‡’   (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œ)))
 
Theoremee13 43866 e13 44110 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 28-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(๐œ‘ โ†’ ๐œ“)    &   (๐œ‘ โ†’ (๐œ’ โ†’ (๐œƒ โ†’ ๐œ)))    &   (๐œ“ โ†’ (๐œ โ†’ ๐œ‚))    โ‡’   (๐œ‘ โ†’ (๐œ’ โ†’ (๐œƒ โ†’ ๐œ‚)))
 
Theoremee121 43867 e121 44018 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(๐œ‘ โ†’ ๐œ“)    &   (๐œ‘ โ†’ (๐œ’ โ†’ ๐œƒ))    &   (๐œ‘ โ†’ ๐œ)    &   (๐œ“ โ†’ (๐œƒ โ†’ (๐œ โ†’ ๐œ‚)))    โ‡’   (๐œ‘ โ†’ (๐œ’ โ†’ ๐œ‚))
 
Theoremee122 43868 e122 44015 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(๐œ‘ โ†’ ๐œ“)    &   (๐œ‘ โ†’ (๐œ’ โ†’ ๐œƒ))    &   (๐œ‘ โ†’ (๐œ’ โ†’ ๐œ))    &   (๐œ“ โ†’ (๐œƒ โ†’ (๐œ โ†’ ๐œ‚)))    โ‡’   (๐œ‘ โ†’ (๐œ’ โ†’ ๐œ‚))
 
Theoremee333 43869 e333 44095 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œƒ)))    &   (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œ)))    &   (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œ‚)))    &   (๐œƒ โ†’ (๐œ โ†’ (๐œ‚ โ†’ ๐œ)))    โ‡’   (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œ)))
 
Theoremee323 43870 e323 44128 without virtual deductions. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œƒ)))    &   (๐œ‘ โ†’ (๐œ“ โ†’ ๐œ))    &   (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œ‚)))    &   (๐œƒ โ†’ (๐œ โ†’ (๐œ‚ โ†’ ๐œ)))    โ‡’   (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œ)))
 
Theorem3ornot23 43871 If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 44209. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((ยฌ ๐œ‘ โˆง ยฌ ๐œ“) โ†’ ((๐œ’ โˆจ ๐œ‘ โˆจ ๐œ“) โ†’ ๐œ’))
 
Theoremorbi1r 43872 orbi1 916 with order of disjuncts reversed. Derived from orbi1rVD 44210. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((๐œ‘ โ†” ๐œ“) โ†’ ((๐œ’ โˆจ ๐œ‘) โ†” (๐œ’ โˆจ ๐œ“)))
 
Theorem3orbi123 43873 pm4.39 975 with a 3-conjunct antecedent. This proof is 3orbi123VD 44212 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((๐œ‘ โ†” ๐œ“) โˆง (๐œ’ โ†” ๐œƒ) โˆง (๐œ โ†” ๐œ‚)) โ†’ ((๐œ‘ โˆจ ๐œ’ โˆจ ๐œ) โ†” (๐œ“ โˆจ ๐œƒ โˆจ ๐œ‚)))
 
Theoremsyl5imp 43874 Closed form of syl5 34. Derived automatically from syl5impVD 44225. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((๐œ‘ โ†’ (๐œ“ โ†’ ๐œ’)) โ†’ ((๐œƒ โ†’ ๐œ“) โ†’ (๐œ‘ โ†’ (๐œƒ โ†’ ๐œ’))))
 
Theoremimpexpd 43875 The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. After the User's Proof was completed, it was minimized. The completed User's Proof before minimization is not shown. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
1:: (((๐œ“ โˆง ๐œ’) โ†’ ๐œƒ) โ†” (๐œ“ โ†’ (๐œ’ โ†’ ๐œƒ)))
qed:1: ((๐œ‘ โ†’ ((๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)) โ†” (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œƒ))))
((๐œ‘ โ†’ ((๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)) โ†” (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œƒ))))
 
Theoremcom3rgbi 43876 The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
1:: ((๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œƒ))) โ†’ (๐œ‘ โ†’ (๐œ’ โ†’ (๐œ“ โ†’ ๐œƒ))))
2:: ((๐œ‘ โ†’ (๐œ’ โ†’ (๐œ“ โ†’ ๐œƒ))) โ†’ (๐œ’ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ ๐œƒ))))
3:1,2: ((๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œƒ))) โ†’ (๐œ’ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ ๐œƒ))))
4:: ((๐œ’ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ ๐œƒ))) โ†’ (๐œ‘ โ†’ (๐œ’ โ†’ (๐œ“ โ†’ ๐œƒ))))
5:: ((๐œ‘ โ†’ (๐œ’ โ†’ (๐œ“ โ†’ ๐œƒ))) โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œƒ))))
6:4,5: ((๐œ’ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ ๐œƒ))) โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œƒ))))
qed:3,6: ((๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œƒ))) โ†” (๐œ’ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ ๐œƒ))))
((๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œƒ))) โ†” (๐œ’ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ ๐œƒ))))
 
Theoremimpexpdcom 43877 The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
1:: ((๐œ‘ โ†’ ((๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)) โ†” (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œƒ))))
2:: ((๐œ“ โ†’ (๐œ’ โ†’ (๐œ‘ โ†’ ๐œƒ))) โ†” (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œƒ))))
qed:1,2: ((๐œ‘ โ†’ ((๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)) โ†” (๐œ“ โ†’ (๐œ’ โ†’ (๐œ‘ โ†’ ๐œƒ))))
((๐œ‘ โ†’ ((๐œ“ โˆง ๐œ’) โ†’ ๐œƒ)) โ†” (๐œ“ โ†’ (๐œ’ โ†’ (๐œ‘ โ†’ ๐œƒ))))
 
Theoremee1111 43878 Non-virtual deduction form of e1111 44037. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
h1:: (๐œ‘ โ†’ ๐œ“)
h2:: (๐œ‘ โ†’ ๐œ’)
h3:: (๐œ‘ โ†’ ๐œƒ)
h4:: (๐œ‘ โ†’ ๐œ)
h5:: (๐œ“ โ†’ (๐œ’ โ†’ (๐œƒ โ†’ (๐œ โ†’ ๐œ‚))))
6:1,5: (๐œ‘ โ†’ (๐œ’ โ†’ (๐œƒ โ†’ (๐œ โ†’ ๐œ‚))))
7:6: (๐œ’ โ†’ (๐œ‘ โ†’ (๐œƒ โ†’ (๐œ โ†’ ๐œ‚))))
8:2,7: (๐œ‘ โ†’ (๐œ‘ โ†’ (๐œƒ โ†’ (๐œ โ†’ ๐œ‚))))
9:8: (๐œ‘ โ†’ (๐œƒ โ†’ (๐œ โ†’ ๐œ‚)))
10:9: (๐œƒ โ†’ (๐œ‘ โ†’ (๐œ โ†’ ๐œ‚)))
11:3,10: (๐œ‘ โ†’ (๐œ‘ โ†’ (๐œ โ†’ ๐œ‚)))
12:11: (๐œ‘ โ†’ (๐œ โ†’ ๐œ‚))
13:12: (๐œ โ†’ (๐œ‘ โ†’ ๐œ‚))
14:4,13: (๐œ‘ โ†’ (๐œ‘ โ†’ ๐œ‚))
qed:14: (๐œ‘ โ†’ ๐œ‚)
(๐œ‘ โ†’ ๐œ“)    &   (๐œ‘ โ†’ ๐œ’)    &   (๐œ‘ โ†’ ๐œƒ)    &   (๐œ‘ โ†’ ๐œ)    &   (๐œ“ โ†’ (๐œ’ โ†’ (๐œƒ โ†’ (๐œ โ†’ ๐œ‚))))    โ‡’   (๐œ‘ โ†’ ๐œ‚)
 
Theorempm2.43bgbi 43879 Logical equivalence of a 2-left-nested implication and a 1-left-nested implicated when two antecedents of the former implication are identical. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
1:: ((๐œ‘ โ†’ (๐œ“ โ†’ (๐œ‘ โ†’ ๐œ’))) โ†’ (๐œ‘ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ ๐œ’))))
2:: ((๐œ‘ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ ๐œ’))) โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ ๐œ’)))
3:1,2: ((๐œ‘ โ†’ (๐œ“ โ†’ (๐œ‘ โ†’ ๐œ’))) โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ ๐œ’)))
4:: ((๐œ‘ โ†’ (๐œ“ โ†’ ๐œ’)) โ†’ (๐œ“ โ†’ (๐œ‘ โ†’ ๐œ’)))
5:3,4: ((๐œ‘ โ†’ (๐œ“ โ†’ (๐œ‘ โ†’ ๐œ’))) โ†’ (๐œ“ โ†’ (๐œ‘ โ†’ ๐œ’)))
6:: ((๐œ“ โ†’ (๐œ‘ โ†’ ๐œ’)) โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ‘ โ†’ ๐œ’))))
qed:5,6: ((๐œ‘ โ†’ (๐œ“ โ†’ (๐œ‘ โ†’ ๐œ’))) โ†” (๐œ“ โ†’ (๐œ‘ โ†’ ๐œ’)))
((๐œ‘ โ†’ (๐œ“ โ†’ (๐œ‘ โ†’ ๐œ’))) โ†” (๐œ“ โ†’ (๐œ‘ โ†’ ๐œ’)))
 
Theorempm2.43cbi 43880 Logical equivalence of a 3-left-nested implication and a 2-left-nested implicated when two antecedents of the former implication are identical. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
1:: ((๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ (๐œ‘ โ†’ ๐œƒ))) ) โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ‘ โ†’ (๐œ’ โ†’ ๐œƒ)))))
2:: ((๐œ‘ โ†’ (๐œ“ โ†’ (๐œ‘ โ†’ (๐œ’ โ†’ ๐œƒ))) ) โ†’ (๐œ“ โ†’ (๐œ‘ โ†’ (๐œ’ โ†’ ๐œƒ))))
3:1,2: ((๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ (๐œ‘ โ†’ ๐œƒ))) ) โ†’ (๐œ“ โ†’ (๐œ‘ โ†’ (๐œ’ โ†’ ๐œƒ))))
4:: ((๐œ“ โ†’ (๐œ‘ โ†’ (๐œ’ โ†’ ๐œƒ))) โ†’ (๐œ“ โ†’ (๐œ’ โ†’ (๐œ‘ โ†’ ๐œƒ))))
5:3,4: ((๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ (๐œ‘ โ†’ ๐œƒ))) ) โ†’ (๐œ“ โ†’ (๐œ’ โ†’ (๐œ‘ โ†’ ๐œƒ))))
6:: ((๐œ“ โ†’ (๐œ’ โ†’ (๐œ‘ โ†’ ๐œƒ))) โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ (๐œ‘ โ†’ ๐œƒ)))))
qed:5,6: ((๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ (๐œ‘ โ†’ ๐œƒ))) ) โ†” (๐œ“ โ†’ (๐œ’ โ†’ (๐œ‘ โ†’ ๐œƒ))))
((๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ (๐œ‘ โ†’ ๐œƒ)))) โ†” (๐œ“ โ†’ (๐œ’ โ†’ (๐œ‘ โ†’ ๐œƒ))))
 
Theoremee233 43881 Non-virtual deduction form of e233 44127. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
h1:: (๐œ‘ โ†’ (๐œ“ โ†’ ๐œ’))
h2:: (๐œ‘ โ†’ (๐œ“ โ†’ (๐œƒ โ†’ ๐œ)))
h3:: (๐œ‘ โ†’ (๐œ“ โ†’ (๐œƒ โ†’ ๐œ‚)))
h4:: (๐œ’ โ†’ (๐œ โ†’ (๐œ‚ โ†’ ๐œ)))
5:1,4: (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ โ†’ (๐œ‚ โ†’ ๐œ))) )
6:5: (๐œ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ‚ โ†’ ๐œ))) )
7:2,6: (๐œ‘ โ†’ (๐œ“ โ†’ (๐œƒ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ‚ โ†’ ๐œ))))))
8:7: (๐œ“ โ†’ (๐œƒ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ‚ โ†’ ๐œ)))))
9:8: (๐œƒ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ‚ โ†’ ๐œ))) )
10:9: (๐œ‘ โ†’ (๐œ“ โ†’ (๐œƒ โ†’ (๐œ‚ โ†’ ๐œ))) )
11:10: (๐œ‚ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ (๐œƒ โ†’ ๐œ))) )
12:3,11: (๐œ‘ โ†’ (๐œ“ โ†’ (๐œƒ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ (๐œƒ โ†’ ๐œ))))))
13:12: (๐œ“ โ†’ (๐œƒ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ (๐œƒ โ†’ ๐œ)))))
14:13: (๐œƒ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ (๐œƒ โ†’ ๐œ))) )
qed:14: (๐œ‘ โ†’ (๐œ“ โ†’ (๐œƒ โ†’ ๐œ)))
(๐œ‘ โ†’ (๐œ“ โ†’ ๐œ’))    &   (๐œ‘ โ†’ (๐œ“ โ†’ (๐œƒ โ†’ ๐œ)))    &   (๐œ‘ โ†’ (๐œ“ โ†’ (๐œƒ โ†’ ๐œ‚)))    &   (๐œ’ โ†’ (๐œ โ†’ (๐œ‚ โ†’ ๐œ)))    โ‡’   (๐œ‘ โ†’ (๐œ“ โ†’ (๐œƒ โ†’ ๐œ)))
 
Theoremimbi13 43882 Join three logical equivalences to form equivalence of implications. imbi13 43882 is imbi13VD 44236 without virtual deductions and was automatically derived from imbi13VD 44236 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((๐œ‘ โ†” ๐œ“) โ†’ ((๐œ’ โ†” ๐œƒ) โ†’ ((๐œ โ†” ๐œ‚) โ†’ ((๐œ‘ โ†’ (๐œ’ โ†’ ๐œ)) โ†” (๐œ“ โ†’ (๐œƒ โ†’ ๐œ‚))))))
 
Theoremee33 43883 Non-virtual deduction form of e33 44096. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
h1:: (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œƒ)))
h2:: (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œ)))
h3:: (๐œƒ โ†’ (๐œ โ†’ ๐œ‚))
4:1,3: (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ (๐œ โ†’ ๐œ‚))))
5:4: (๐œ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œ‚))))
6:2,5: (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œ‚))))))
7:6: (๐œ“ โ†’ (๐œ’ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œ‚)))))
8:7: (๐œ’ โ†’ (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œ‚))))
qed:8: (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œ‚)))
(๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œƒ)))    &   (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œ)))    &   (๐œƒ โ†’ (๐œ โ†’ ๐œ‚))    โ‡’   (๐œ‘ โ†’ (๐œ“ โ†’ (๐œ’ โ†’ ๐œ‚)))
 
Theoremcon5 43884 Biconditional contraposition variation. This proof is con5VD 44262 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((๐œ‘ โ†” ยฌ ๐œ“) โ†’ (ยฌ ๐œ‘ โ†’ ๐œ“))
 
Theoremcon5i 43885 Inference form of con5 43884. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(๐œ‘ โ†” ยฌ ๐œ“)    โ‡’   (ยฌ ๐œ‘ โ†’ ๐œ“)
 
Theoremexlimexi 43886 Inference similar to Theorem 19.23 of [Margaris] p. 90. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(๐œ“ โ†’ โˆ€๐‘ฅ๐œ“)    &   (โˆƒ๐‘ฅ๐œ‘ โ†’ (๐œ‘ โ†’ ๐œ“))    โ‡’   (โˆƒ๐‘ฅ๐œ‘ โ†’ ๐œ“)
 
Theoremsb5ALT 43887* Equivalence for substitution. Alternate proof of sb5 2260. This proof is sb5ALTVD 44275 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
([๐‘ฆ / ๐‘ฅ]๐œ‘ โ†” โˆƒ๐‘ฅ(๐‘ฅ = ๐‘ฆ โˆง ๐œ‘))
 
Theoremeexinst01 43888 exinst01 43987 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
โˆƒ๐‘ฅ๐œ“    &   (๐œ‘ โ†’ (๐œ“ โ†’ ๐œ’))    &   (๐œ‘ โ†’ โˆ€๐‘ฅ๐œ‘)    &   (๐œ’ โ†’ โˆ€๐‘ฅ๐œ’)    โ‡’   (๐œ‘ โ†’ ๐œ’)
 
Theoremeexinst11 43889 exinst11 43988 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(๐œ‘ โ†’ โˆƒ๐‘ฅ๐œ“)    &   (๐œ‘ โ†’ (๐œ“ โ†’ ๐œ’))    &   (๐œ‘ โ†’ โˆ€๐‘ฅ๐œ‘)    &   (๐œ’ โ†’ โˆ€๐‘ฅ๐œ’)    โ‡’   (๐œ‘ โ†’ ๐œ’)
 
Theoremvk15.4j 43890 Excercise 4j of Unit 15 of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia Klenk. This proof is the minimized Hilbert-style axiomatic version of the Fitch-style Natural Deduction proof found on page 442 of Klenk and was automatically derived from that proof. vk15.4j 43890 is vk15.4jVD 44276 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
ยฌ (โˆƒ๐‘ฅ ยฌ ๐œ‘ โˆง โˆƒ๐‘ฅ(๐œ“ โˆง ยฌ ๐œ’))    &   (โˆ€๐‘ฅ๐œ’ โ†’ ยฌ โˆƒ๐‘ฅ(๐œƒ โˆง ๐œ))    &    ยฌ โˆ€๐‘ฅ(๐œ โ†’ ๐œ‘)    โ‡’   (ยฌ โˆƒ๐‘ฅ ยฌ ๐œƒ โ†’ ยฌ โˆ€๐‘ฅ๐œ“)
 
TheoremnotnotrALT 43891 Converse of double negation. Alternate proof of notnotr 130. This proof is notnotrALTVD 44277 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(ยฌ ยฌ ๐œ‘ โ†’ ๐œ‘)
 
Theoremcon3ALT2 43892 Contraposition. Alternate proof of con3 153. This proof is con3ALTVD 44278 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((๐œ‘ โ†’ ๐œ“) โ†’ (ยฌ ๐œ“ โ†’ ยฌ ๐œ‘))
 
Theoremssralv2 43893* Quantification restricted to a subclass for two quantifiers. ssralv 4046 for two quantifiers. The proof of ssralv2 43893 was automatically generated by minimizing the automatically translated proof of ssralv2VD 44228. The automatic translation is by the tools program translate_without_overwriting.cmd. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((๐ด โІ ๐ต โˆง ๐ถ โІ ๐ท) โ†’ (โˆ€๐‘ฅ โˆˆ ๐ต โˆ€๐‘ฆ โˆˆ ๐ท ๐œ‘ โ†’ โˆ€๐‘ฅ โˆˆ ๐ด โˆ€๐‘ฆ โˆˆ ๐ถ ๐œ‘))
 
Theoremsbc3or 43894 sbcor 3827 with a 3-disjuncts. This proof is sbc3orgVD 44213 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Revised by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
([๐ด / ๐‘ฅ](๐œ‘ โˆจ ๐œ“ โˆจ ๐œ’) โ†” ([๐ด / ๐‘ฅ]๐œ‘ โˆจ [๐ด / ๐‘ฅ]๐œ“ โˆจ [๐ด / ๐‘ฅ]๐œ’))
 
Theoremalrim3con13v 43895* Closed form of alrimi 2199 with 2 additional conjuncts having no occurrences of the quantifying variable. This proof is 19.21a3con13vVD 44214 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((๐œ‘ โ†’ โˆ€๐‘ฅ๐œ‘) โ†’ ((๐œ“ โˆง ๐œ‘ โˆง ๐œ’) โ†’ โˆ€๐‘ฅ(๐œ“ โˆง ๐œ‘ โˆง ๐œ’)))
 
Theoremrspsbc2 43896* rspsbc 3869 with two quantifying variables. This proof is rspsbc2VD 44217 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(๐ด โˆˆ ๐ต โ†’ (๐ถ โˆˆ ๐ท โ†’ (โˆ€๐‘ฅ โˆˆ ๐ต โˆ€๐‘ฆ โˆˆ ๐ท ๐œ‘ โ†’ [๐ถ / ๐‘ฆ][๐ด / ๐‘ฅ]๐œ‘)))
 
Theoremsbcoreleleq 43897* Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 44221. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(๐ด โˆˆ ๐‘‰ โ†’ ([๐ด / ๐‘ฆ](๐‘ฅ โˆˆ ๐‘ฆ โˆจ ๐‘ฆ โˆˆ ๐‘ฅ โˆจ ๐‘ฅ = ๐‘ฆ) โ†” (๐‘ฅ โˆˆ ๐ด โˆจ ๐ด โˆˆ ๐‘ฅ โˆจ ๐‘ฅ = ๐ด)))
 
Theoremtratrb 43898* If a class is transitive and any two distinct elements of the class are E-comparable, then every element of that class is transitive. Derived automatically from tratrbVD 44223. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((Tr ๐ด โˆง โˆ€๐‘ฅ โˆˆ ๐ด โˆ€๐‘ฆ โˆˆ ๐ด (๐‘ฅ โˆˆ ๐‘ฆ โˆจ ๐‘ฆ โˆˆ ๐‘ฅ โˆจ ๐‘ฅ = ๐‘ฆ) โˆง ๐ต โˆˆ ๐ด) โ†’ Tr ๐ต)
 
TheoremordelordALT 43899 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 6385 using the Axiom of Regularity indirectly through dford2 9635. dford2 is a weaker definition of ordinal number. Given the Axiom of Regularity, it need not be assumed that E Fr ๐ด because this is inferred by the Axiom of Regularity. ordelordALT 43899 is ordelordALTVD 44229 without virtual deductions and was automatically derived from ordelordALTVD 44229 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((Ord ๐ด โˆง ๐ต โˆˆ ๐ด) โ†’ Ord ๐ต)
 
Theoremsbcim2g 43900 Distribution of class substitution over a left-nested implication. Similar to sbcimg 3825. sbcim2g 43900 is sbcim2gVD 44237 without virtual deductions and was automatically derived from sbcim2gVD 44237 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(๐ด โˆˆ ๐‘‰ โ†’ ([๐ด / ๐‘ฅ](๐œ‘ โ†’ (๐œ“ โ†’ ๐œ’)) โ†” ([๐ด / ๐‘ฅ]๐œ‘ โ†’ ([๐ด / ๐‘ฅ]๐œ“ โ†’ [๐ด / ๐‘ฅ]๐œ’))))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48161
  Copyright terms: Public domain < Previous  Next >