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Theorem List for Metamath Proof Explorer - 24901-25000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
18.11  Mathbox for Wolf Lammen

Most of the theorems in the section "Logical implication" are about handling chains of implications:  ph  ->  ( ps  ->  ( ch  ->  .... With respect to chains, an rich set of rules clarify

- how to swap antecedents (com12, ...);

- how to drop antecedents (ax-mp, pm2.43, ...);

- how to add antecedents (a1i, ...)

- how to replace an antecedent (syl, ...);

- how to replace a consequent (ax-mp, syl, ...);

- what is, when an antecedent equals the consequent (ax-1, id, ...).

In all these cases, the operands of the chain have no inner structure, or it is of no importance. These chains are called "simple" here.

There is less support, when the operands are structured themselves. Some kinds of inner structure involving the  -. operator are best handled by the symmetric operators  /\ and  \/. But a nested, simple chain has no such convenient replacement. I can focus on antecedents here, since a consequent representing a chain is, in conjunction with its antecedents, just an extended simple chain again.

The following theorems show, how operations on nested chains appear somehow mirrored: The minor premises of the syllogisms look reverted, in comparison to their normal counterparts, and while adding an antecedent to a chain via a1i 10 is easy, in nested chains they can be easily dropped.

 
Theoremwl-jarri 24901 Dropping a nested antecedent. This theorem is one of two reversions of ja 153. Since ja 153 is reversible, one can conclude, that a nested (chain of) implication(s) is just a packed notation of two or more theorems/ hypotheses with a common consequent. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

ax46 2101 is an instance of this idea.

 |-  (
 ( ph  ->  ps )  ->  ch )   =>    |-  ( ps  ->  ch )
 
Theoremwl-jarli 24902 Dropping a nested consequent. This theorem is one of two reversions of ja 153. Since ja 153 is reversible, one can conclude, that a nested (chain of) implication(s) is just a packed notation of two or more theorems/ hypotheses with a common consequent. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

ax46 2101 is an instance of this idea.

 |-  (
 ( ph  ->  ps )  ->  ch )   =>    |-  ( -.  ph  ->  ch )
 
Theoremwl-mps 24903 Replacing a nested consequent. A sort of modus ponens in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ( ph  ->  ch )  ->  th )   =>    |-  (
 ( ph  ->  ps )  ->  th )
 
Theoremwl-syls1 24904 Replacing a nested consequent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ps  ->  ch )   &    |-  ( ( ph  ->  ch )  ->  th )   =>    |-  (
 ( ph  ->  ps )  ->  th )
 
Theoremwl-syls2 24905 Replacing a nested antecedent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ( ph  ->  ch )  ->  th )   =>    |-  (
 ( ps  ->  ch )  ->  th )
 
Theoremwl-adnestant 24906 A true wff can always be added as a nested antecedent to an antecedent. Note: this theorem is intuitionistically valid (see wl-adnestantALT 24907) (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ch )   =>    |-  ( ( ph  ->  ps )  ->  ch )
 
Theoremwl-adnestantALT 24907 Proof of wl-adnestant 24906 not based on ax-3 7. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ch )   =>    |-  ( ( ph  ->  ps )  ->  ch )
 
Theoremwl-adnestantd 24908 Deduction version of wl-adnestant 24906. Generalization of a2i 12, imim12i 53, imim1i 54 and imim2i 13, which can be proved by specializing its hypotheses, and some trivial rearrangements. This theorem clarifies in a more general way, under what conditions a wff may be introduced as a nested antecedent to an antecedent. Note: this theorem is intuitionistically valid (see wl-adnestantdALTOLD 24909). (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th )
 )   =>    |-  ( ph  ->  (
 ( ps  ->  ch )  ->  th ) )
 
Theoremwl-adnestantdALTOLD 24909 Proof of wl-adnestantd 24908 not based on ax-3 7. (Contributed by Wolf Lammen, 4-Oct-2013.) (Moved to embantd 50 in main set.mm and may be deleted by mathbox owner, WL. --NM 14-Jan-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th )
 )   =>    |-  ( ph  ->  (
 ( ps  ->  ch )  ->  th ) )
 
Theoremwl-bitr1 24910 Closed form of bitri 240. Place before bitri 240. [ +33] (Contributed by Wolf Lammen, 5-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ps  <->  ch )  ->  ( ph 
 <->  ch ) ) )
 
Theoremwl-bitri 24911 An inference from transitive law for logical equivalence. [ -5] (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph 
 <->  ps )   &    |-  ( ps  <->  ch )   =>    |-  ( ph  <->  ch )
 
Theoremwl-bitrd 24912 Deduction form of bitri 240. [ -7] (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  <->  ch ) )   &    |-  ( ph  ->  ( ch  <->  th ) )   =>    |-  ( ph  ->  ( ps  <->  th ) )
 
Theoremwl-bibi1 24913 Theorem *4.86 of [WhiteheadRussell] p. 122. Place this (and the following theorems) after bitr1. [ +22] (Contributed by NM, 3-Jan-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ph  <->  ch )  <->  ( ps  <->  ch ) ) )
 
Theoremwl-bibi1i 24914 Inference adding a biconditional to the right in an equivalence. Move after bibi1. [ -8] (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph 
 <->  ps )   =>    |-  ( ( ph  <->  ch )  <->  ( ps  <->  ch ) )
 
Theoremwl-bibi1d 24915 Deduction adding a biconditional to the right in an equivalence. Move after bibi1i. [ -9] (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( ( ps  <->  th )  <->  ( ch  <->  th ) ) )
 
Theoremwl-bibi2d 24916 Deduction adding a biconditional to the left in an equivalence. Move after bibi1d. [ -25] (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( ( th  <->  ps )  <->  ( th  <->  ch ) ) )
 
Theoremwl-pm5.74lem 24917 Moving a common antecedent on one side of an equivalence. Place before pm5.74 235. [ +25] (Contributed by Wolf Lammen, 5-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  <->  ch ) )   &    |-  ( -.  ph  ->  ch )   =>    |-  (
 ( ph  ->  ps )  <->  ch )
 
Theoremwl-pm5.74 24918 Distribution of implication over biconditional. Theorem *5.74 of [ WhiteheadRussell] p. 126. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 11-Apr-2013.) Replace and move biimt 325.. albi 1551 before it. [ -22] (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps  <->  ch ) )  <->  ( ( ph  ->  ps )  <->  ( ph  ->  ch ) ) )
 
Theoremwl-pm5.32 24919 Distribution of implication over biconditional. Theorem *5.32 of [ WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 5-Oct-2013.) Replace. [ -43] (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps  <->  ch ) )  <->  ( ( ph  /\ 
 ps )  <->  ( ph  /\  ch ) ) )
 
Theoremwl-bitr 24920 Theorem *4.22 of [WhiteheadRussell] p. 117. Replace. [ -4] (Contributed by NM, 3-Jan-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ps 
 <->  ch ) )  ->  ( ph  <->  ch ) )
 
Theoremwl-pm2.86i 24921 Inference based on pm2.86 94. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ps )  ->  ( ph  ->  ch )
 )   =>    |-  ( ph  ->  ( ps  ->  ch ) )
 
Theoremwl-dedlem0a 24922 Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  <->  ( ( ch 
 ->  ph )  ->  ( ps  /\  ph ) ) ) )
 
18.12  Mathbox for Brendan Leahy
 
Theoremdvreasin 24923 Real derivative of arcsine. (Contributed by Brendan Leahy, 3-Aug-2017.)
 |-  ( RR  _D  (arcsin  |`  ( -u 1 (,) 1 ) ) )  =  ( x  e.  ( -u 1 (,) 1 )  |->  ( 1 
 /  ( sqr `  (
 1  -  ( x ^ 2 ) ) ) ) )
 
Theoremdvreacos 24924 Real derivative of arccosine. (Contributed by Brendan Leahy, 3-Aug-2017.)
 |-  ( RR  _D  (arccos  |`  ( -u 1 (,) 1 ) ) )  =  ( x  e.  ( -u 1 (,) 1 )  |->  ( -u 1  /  ( sqr `  (
 1  -  ( x ^ 2 ) ) ) ) )
 
Theoremareacirclem2 24925* Antiderivative of cross-section of circle. (Contributed by Brendan Leahy, 28-Aug-2017.)
 |-  ( R  e.  RR+  ->  ( RR  _D  ( t  e.  ( -u R (,) R )  |->  ( ( R ^ 2 )  x.  ( (arcsin `  (
 t  /  R )
 )  +  ( ( t  /  R )  x.  ( sqr `  (
 1  -  ( ( t  /  R ) ^ 2 ) ) ) ) ) ) ) )  =  ( t  e.  ( -u R (,) R )  |->  ( 2  x.  ( sqr `  ( ( R ^
 2 )  -  (
 t ^ 2 ) ) ) ) ) )
 
Theoremareacirclem3 24926* Continuity of cross-section of circle. (Contributed by Brendan Leahy, 28-Aug-2017.)
 |-  ( R  e.  RR+  ->  (
 t  e.  ( -u R (,) R )  |->  ( 2  x.  ( sqr `  ( ( R ^
 2 )  -  (
 t ^ 2 ) ) ) ) )  e.  ( ( -u R (,) R ) -cn-> CC ) )
 
Theoremareacirclem4 24927* Endpoint-inclusive continuity of Cartesian ordinate of circle. (Contributed by Brendan Leahy, 29-Aug-2017.)
 |-  (
 ( R  e.  RR  /\  0  <_  R )  ->  ( t  e.  ( -u R [,] R ) 
 |->  ( sqr `  (
 ( R ^ 2
 )  -  ( t ^ 2 ) ) ) )  e.  (
 ( -u R [,] R ) -cn-> CC ) )
 
Theoremareacirclem1 24928* Integrability of cross-section of circle. (Contributed by Brendan Leahy, 26-Aug-2017.)
 |-  (
 ( R  e.  RR  /\  0  <_  R )  ->  ( t  e.  ( -u R [,] R ) 
 |->  ( 2  x.  ( sqr `  ( ( R ^ 2 )  -  ( t ^ 2
 ) ) ) ) )  e.  L ^1 )
 
Theoremareacirclem5 24929* Endpoint-inclusive continuity of antiderivative of cross-section of circle. (Contributed by Brendan Leahy, 31-Aug-2017.)
 |-  ( R  e.  RR+  ->  (
 t  e.  ( -u R [,] R )  |->  ( ( R ^ 2
 )  x.  ( (arcsin `  ( t  /  R ) )  +  (
 ( t  /  R )  x.  ( sqr `  (
 1  -  ( ( t  /  R ) ^ 2 ) ) ) ) ) ) )  e.  ( (
 -u R [,] R ) -cn-> CC ) )
 
Theoremareacirclem6 24930* Finding the cross-section of a circle. (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 22-Sep-2017.)
 |-  S  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  (
 ( x ^ 2
 )  +  ( y ^ 2 ) ) 
 <_  ( R ^ 2
 ) ) }   =>    |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e. 
 RR )  ->  ( S " { t }
 )  =  if (
 ( abs `  t )  <_  R ,  ( -u ( sqr `  ( ( R ^ 2 )  -  ( t ^ 2
 ) ) ) [,] ( sqr `  (
 ( R ^ 2
 )  -  ( t ^ 2 ) ) ) ) ,  (/) ) )
 
Theoremareacirc 24931* The area of a circle of radius  R is  pi  x.  R ^ 2. (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 22-Sep-2017.)
 |-  S  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  (
 ( x ^ 2
 )  +  ( y ^ 2 ) ) 
 <_  ( R ^ 2
 ) ) }   =>    |-  ( ( R  e.  RR  /\  0  <_  R )  ->  (area `  S )  =  ( pi  x.  ( R ^ 2 ) ) )
 
18.13  Mathbox for Frédéric Liné

In the sequel "JFM" is the "Journal of Formalized Mathematics". http://mizar.uwb.edu.pl/JFM/mmlident.html

"CAT1"; means Bylinski Czeslaw, Introduction to Categories and Functors, Journal of Formalized Mathematics, 1990, volume 1, no 2, pages 409--420

"CAT2"; means Bylinski Czeslaw, Subcategories and Products of Categories, Journal of Formalized Mathematics, 1990, volume 1, no 4, pages 725--732

"CLASSES1" means Grzegorz Bancerek, Tarski's Classes and Ranks, Journal of Formalized Mathematics, 1990, volume 1, no 3, pages 563--567

"CLASSES2" means Bogdan Nowak and Grzegorz Bancerek, Universal Classes, Journal of Formalized Mathematics, may-august 1990, volume 1, nb 3, pages 595--600

"Bourbaki" means Bourbaki's treatise. The book General Topology is called TG (for Topologie Générale). The book Set Theory is called E (for théorie des Ensembles).

The treatise is translated in English.

More precisely, here are two examples of references:

"Bourbaki E II.32" means Set Theory, chapter II, 32nd page, "Bourbaki TG III.1" means General Topology, chapter III, 1st page.

The references are given according to the French edition.

"Viro" means O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, V. M. Kharlamov, Elementary topology. Available on the net.

"Goldblatt" means Robert Goldblatt, Topoi, the categorial analysis of logic, revised edition, Dover publications, Mineola, New-York, 2006

"Gallier" means Jean H. Gallier, "Logic For Computer Science -- Foundations of Automatic Theorem Proving". A new edition must be published in 2014 at Dover.

"Harju" means Tero Harju, "Lecture Notes on SemiGroups", unpublished, 1996. Available on the net.

In the following notices "experimental" means I have not yet sufficiently used a definition to be sure it is correct. Anyway I'm not the owner of the definition and you can use it as you wish if you think it is correct or replace it by a definition of your own if you think it is not.

 
18.13.1  Theorems from other workspaces
 
Theoremtpssg 24932 A triplet of elements of a class is a subset of the class. (Contributed by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Revised by FL, 17-May-2016.)
 |-  ( ph  ->  A  e.  E )   &    |-  ( ph  ->  B  e.  F )   &    |-  ( ph  ->  C  e.  G )   =>    |-  ( ph  ->  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  <->  { A ,  B ,  C }  C_  D ) )
 
18.13.2  Propositional and predicate calculus
 
Theoremneleq12d 24933 Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  e/  C  <->  B  e/  D ) )
 
Theoremr19.26-2a 24934 Theorem 19.26 of [Margaris] p. 90 with 3 restricted quantifiers. (Contributed by FL, 20-May-2016.)
 |-  ( A. x  e.  A  A. y  e.  B  A. z  e.  C  ( ph  /\  ps )  <->  ( A. x  e.  A  A. y  e.  B  A. z  e.  C  ph  /\  A. x  e.  A  A. y  e.  B  A. z  e.  C  ps ) )
 
Theoremreubidvag 24935* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by FL, 17-Nov-2014.)
 |-  ( ph  ->  ( ps  <->  ch ) )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( E! x  e.  A  ps 
 <->  E! x  e.  B  ch ) )
 
Theoremintn3an1d 24936 Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  -.  ( ps  /\ 
 ch  /\  th )
 )
 
Theoremintn3an2d 24937 Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  -.  ( ch  /\ 
 ps  /\  th )
 )
 
Theoremintn3an3d 24938 Introduction of a triple conjunct inside a contradiction. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  -.  ( ch  /\ 
 th  /\  ps )
 )
 
Theoremand4as 24939 A consequence of  /\ associativity in a triple conjunct. (Contributed by FL, 14-Jul-2007.)
 |-  (
 ( ph  /\  ps  /\  ( ch  /\  th )
 ) 
 <->  ( ( ph  /\  ps  /\ 
 ch )  /\  th ) )
 
Theoremand4com 24940 A consequence of  /\ associativity in a triple conjunct. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  (
 ( ph  /\  ( ps 
 /\  ch  /\  th )
 ) 
 <->  ( ( ph  /\  ps  /\ 
 ch )  /\  th ) )
 
Theoremanddi2 24941 Conjunction of triple disjunctions. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  (
 ( ( ph  \/  ps 
 \/  ch )  /\  ( th  \/  ta  \/  et ) )  <->  ( ( (
 ph  /\  th )  \/  ( ph  /\  ta )  \/  ( ph  /\  et ) )  \/  (
 ( ps  /\  th )  \/  ( ps  /\  ta )  \/  ( ps 
 /\  et ) )  \/  ( ( ch  /\  th )  \/  ( ch 
 /\  ta )  \/  ( ch  /\  et ) ) ) )
 
Theoremcondis 24942 Proof by contradiction combined with a disjunction. (Contributed by FL, 20-Apr-2011.)
 |-  ( ph  ->  ps )   &    |-  ( -.  ph  ->  ch )   =>    |-  ( ps  \/  ch )
 
Theoremcondisd 24943 Proof by contradiction combined with a disjunction. (Contributed by FL, 20-Apr-2011.)
 |-  (
 ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 -.  ps )  ->  th )   =>    |-  ( ph  ->  ( ch  \/  th ) )
 
Theoremeeeeanv 24944* Rearrange existential quantifiers. (Contributed by FL, 14-Jul-2007.)
 |-  ( E. w E. x E. y E. z ( (
 ph  /\  ps  /\  ch )  /\  th )  <->  ( ( E. w ph  /\  E. x ps  /\  E. y ch )  /\  E. z th ) )
 
Theoremnegcmpprcal1 24945 Negation of a complex predicate calculus formula. (Contributed by FL, 31-Jul-2009.)
 |-  ( -.  E. x  e.  A  A. y  e.  B  (
 ph  ->  ps )  <->  A. x  e.  A  E. y  e.  B  ( ph  /\  -.  ps ) )
 
Theoremnegcmpprcal2 24946 Negation of a complex predicated inequality. (Contributed by FL, 31-Jul-2009.)
 |-  ( -.  E. x  e.  A  A. y  e.  B  C  =/=  D  <->  A. x  e.  A  E. y  e.  B  C  =  D )
 
Theoremeqriv2 24947 Infer equality of classes from equivalence of membership. (Contributed by FL, 22-Nov-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  ( x  e.  A  <->  x  e.  B )   =>    |-  A  =  B
 
Theoremaltdftru 24948 Alternate definition of true. In fact any tautology is a definition of true. (Contributed by FL, 23-Mar-2011.)
 |-  (  T. 
 <->  ( ph  \/  -.  ph ) )
 
Theoremtrant 24949 A true antecedent can be removed. (Contributed by FL, 16-Apr-2012.)
 |-  (
 (  T.  ->  ph )  <->  ph )
 
Theoremvutr 24950 Vacuous universal quantification is true. (Contributed by FL, 16-Apr-2012.)
 |-  (  T. 
 <-> 
 A. x  e.  (/)  ph )
 
Theoremtrcrm 24951 True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.)
 |-  (
 (  T.  /\  ph )  <->  ph )
 
Theoremtnf 24952 True is not false. (Contributed by FL, 20-Mar-2011.)
 |-  (  T. 
 <->  -.  F.  )
 
Theoremfacrm 24953 False can be removed from a disjunction. (Contributed by FL, 20-Mar-2011.)
 |-  (
 (  F.  \/  ph )  <->  ph )
 
Theoremfordisxex 24954 If  ( ph  \/  ps ) is true for all  x and  ps is not true for all  x then  ph is true for some  x. (Contributed by FL, 20-Apr-2011.)
 |-  (
 ( A. x  e.  A  ( ph  \/  ps )  /\  -.  A. x  e.  A  ps )  ->  E. x  e.  A  ph )
 
Theoremfates 24955* Equivalence of  A. and  E. in the case of quantifiers restricted to a singleton. (Contributed by FL, 1-Jun-2011.)
 |-  A  e.  B   =>    |-  ( A. x  e. 
 { A } ph  <->  E. x  e.  { A } ph )
 
Theoremfatesg 24956* Equivalence of  A. and  E. in the case of quantifiers restricted to a singleton. (Contributed by FL, 1-Jun-2011.)
 |-  ( A  e.  V  ->  (
 A. x  e.  { A } ph  <->  E. x  e.  { A } ph ) )
 
Theoremr19.2zr 24957* Quantifying a hypothesis with a universal restricted quantifier. (Contributed by FL, 19-Sep-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ph )  ->  ps )
 
Theoremr19.2zrr 24958* Removing a universal restricted quantifier when the variable doesn't occur in the proposition. (Contributed by FL, 19-Sep-2011.)
 |-  (
 ( A  =/=  (/)  /\  A. x  e.  A  ph )  -> 
 ph )
 
Theoremrexlimib 24959* Removal of an universal restricted quantifier in an antecedent. See also reximdva0 3466. (Contributed by FL, 19-Apr-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/ x ps   &    |-  ( x  e.  A  ->  ( ph  ->  ps ) )   =>    |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ph )  ->  ps )
 
Theoremeqint 24960* To prove that a set  A is the finest one that has the property  ph, prove that  A is a part of all sets that has this property and that  A has also that property. (Contributed by FL, 21-Apr-2012.)
 |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  ps   &    |-  ( ph  ->  A  C_  x )   =>    |-  ( A  e.  V  ->  A  =  |^| { x  |  ph } )
 
Theoremeqintg 24961* To prove that a set  A is the finest one that has the property  ph prove that  A is a part of all sets that has this property and that  A has also that property. (Contributed by FL, 15-Oct-2012.)
 |-  ( x  =  A  ->  ( ch  <->  ps ) )   &    |-  ( ph  ->  ch )   &    |-  ( ( ph  /\ 
 ps )  ->  A  C_  x )   =>    |-  ( ( ph  /\  A  e.  V )  ->  A  =  |^| { x  |  ps } )
 
Theoremalexeqd 24962* Two ways to express substitution of 
A for  x in  ph. (Contributed by FL, 4-Jun-2012.)
 |-  ( A  e.  V  ->  (
 A. x ( x  =  A  ->  ph )  <->  E. x ( x  =  A  /\  ph )
 ) )
 
Theoremrspc2edv 24963* 2-variable restricted existential specialization, using implicit substitution. (rspc2ev 2892 with an antecedent.) (Contributed by FL, 2-Jul-2012.)
 |-  ( x  =  A  ->  ( ps  <->  th ) )   &    |-  (
 y  =  B  ->  ( th  <->  ch ) )   =>    |-  ( ( ph  /\  A  e.  C  /\  B  e.  D )  ->  ( ch  ->  E. x  e.  C  E. y  e.  D  ps ) )
 
Theorempm11.53g 24964 Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by FL, 27-Oct-2013.)
 |-  F/ y ph   &    |-  F/ x ps   =>    |-  ( A. x A. y (
 ph  ->  ps )  <->  ( E. x ph 
 ->  A. y ps )
 )
 
Theoremeqvinopb 24965* A variable introduction law for ordered triples. See eqvinop 4251. (Contributed by FL, 6-Nov-2013.)
 |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( A  =  <. <. B ,  C >. ,  D >.  <->  E. x E. y E. z ( A  =  <.
 <. x ,  y >. ,  z >.  /\  <. <. x ,  y >. ,  z >.  = 
 <. <. B ,  C >. ,  D >. ) )
 
Theoremcopsexgb 24966* Substitution of class  A for ordered triple  <. <. x ,  y >. ,  z
>.. See copsexg 4252. (Contributed by FL, 10-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  ( A  =  <. <. x ,  y >. ,  z >.  ->  ( ph  <->  E. x E. y E. z ( A  =  <.
 <. x ,  y >. ,  z >.  /\  ph )
 ) )
 
Theoremdifeq12dOLD 24967 Deduction adding difference to the right in a class equality. (Moved into main set.mm as difeq12d 3295 and may be deleted by mathbox owner, FL. --NM 2-Jul-2014.) (Contributed by FL, 29-May-2014.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  \  C )  =  ( B  \  D ) )
 
Theorem3netr3 24968 Inequality. (Contributed by FL, 30-May-2014.)
 |-  A  =/=  B   &    |-  A  =  C   &    |-  B  =  D   =>    |-  C  =/=  D
 
Theoremsbcbidv2 24969* Formula-building deduction rule for class substitution with different classes. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 [. A  /  x ].
 ps 
 <-> 
 [. B  /  x ].
 ch ) )
 
18.13.3  Linear temporal logic

Propositional Linear temporal logic (LTL) is a kind of modal logic. It is composed of the axioms of classical logic plus the axioms ax-ltl1 24974, ax-ltl2 24975, ax-ltl3 24976, ax-ltl4 , ax-lmp 24978, and ax-nmp 24979. In classical logic, propositions don't depend on the time. In LTL the "world" evolves. We will imagine the world as a sequence of states with a first state and future states. Instead of state I will also use the term "step" to emphasize that LTL is used to formalize the evolution of process in a computer. A proposition that is true in one state of the "world" may be false in the next one. The proposition  [.] ph means  ph is true in every state of the world, in the first state as well as in the future states. It is read "
ph is always true " or " ph always holds ". The proposition  () ph means  ph is true in the next state of the world. The proposition 
<> ph means that  ph is true in one state of the world at least but we don't know exactly which one. It can be the first state, it can be a future state. It is read " ph is eventually true " or " ph eventually holds". When no operator is used in front of a proposition, it means that  ph is unconditionnaly true or that it is true in the current state ( depending on the context).  ph  until  ps means  ph is true in every state of the world until  ps is true.

 
Syntaxwbox 24970 An always true proposition is well formed.
 wff  [.] ph
 
Syntaxwdia 24971 An eventually true proposition is well formed.
 wff  <> ph
 
Syntaxwcirc 24972 A proposition true in the next step is well formed.
 wff  ()
 ph
 
Syntaxwunt 24973 The proposition " ph is true until  ps is true " is well formed.
 wff  ( ph  until  ps )
 
Axiomax-ltl1 24974 If  ( ph  ->  ps ) and  ph always hold then  ps always holds. (Contributed by FL, 22-Feb-2011.)
 |-  ( [.] ( ph  ->  ps )  ->  ( [.] ph  ->  [.]
 ps ) )
 
Axiomax-ltl2 24975  ph doesn't hold in the next step iff in the next step 
-.  ph holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( -.  () ph  <->  ()  -.  ph )
 
Axiomax-ltl3 24976 If, in the next step,  ph  ->  ps and  ph hold then, in the next step,  ps holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( () ( ph  ->  ps )  ->  ( () ph  ->  ()
 ps ) )
 
Axiomax-ltl4 24977 Suppose that it is always true that if  ph is true in the current step then  ph is true in the next step. Suppose that  ph is true in the first step. Then  ph is always true. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( [.] ( ph  ->  ()
 ph )  /\  ph )  ->  [.] ph )
 
Axiomax-lmp 24978 If  ph is a theorem then it always holds. (Contributed by FL, 22-Feb-2011.)
 |-  ph   =>    |- 
 [.] ph
 
Axiomax-nmp 24979 If  ph is a theorem then it holds in the next step. (Contributed by FL, 20-Mar-2011.)
 |-  ph   =>    |- 
 () ph
 
Definitiondf-dia 24980  ph eventually holds iff it is not true that  -.  ph always holds. (Contributed by FL, 22-Feb-2011.)
 |-  ( <> ph  <->  -. 
 [.]  -.  ph )
 
Theoremimpbox 24981 If  ph  ->  ps is unconditionally true and if  ph is always true then  ps is always true. (Contributed by FL, 22-Feb-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( [.] ph  ->  [.]
 ps )
 
Theorembibox 24982 If  ph  <->  ps is unconditionally true then  ph is always true is equivalent to  ps is always true. (Contributed by FL, 22-Feb-2011.)
 |-  ( ph 
 <->  ps )   =>    |-  ( [.] ph  <->  [.] ps )
 
Theoremimpxt 24983 If  ph  ->  ps holds unconditionally and if  ph holds in the next state then  ps holds in the next state. (Contributed by FL, 20-Mar-2011.)
 |-  ( ph  ->  ps )   =>    |-  ( () ph  ->  ()
 ps )
 
Theorembinxt 24984 If  ph  <->  ps holds unconditionally then  ph holds in the next state of the world iff  ps holds in the next state. (Contributed by FL, 20-Mar-2011.)
 |-  ( ph 
 <->  ps )   =>    |-  ( () ph  <->  () ps )
 
Theoremnxtor 24985  ( ph  \/  ps ) holds in the next step iff  ph holds in the next step or  ps holds in the next step. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( () ( ph  \/  ps ) 
 <->  ( () ph  \/  () ps ) )
 
Theoremnxtand 24986  ( ph  /\ 
ps ) holds in the next step iff  ph holds in the next step and  ps holds in the next step. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( () ( ph  /\  ps ) 
 <->  ( () ph  /\  () ps ) )
 
Theoremboxeq 24987  ph holds now and will always hold in the future iff it is not true that  -.  ph holds now or sometimes in the future. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( [.] ph  <->  -.  <>  -.  ph )
 
Theoremdiaimi 24988 If  ph implies  ps unconditionally, then if  ph eventually holds so does  ps. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( ph  ->  ps )   =>    |-  ( <> ph  ->  <> ps )
 
Theorembidia 24989 If  ph  <->  ps holds then  ph eventually holds iff  ps eventually holds. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Mario Carneiro, 30-Aug-2016.)
 |-  ( ph 
 <->  ps )   =>    |-  ( <> ph  <->  <> ps )
 
Theoremnotev 24990 It's false that  ph eventually holds iff  -.  ph always holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( -. 
 <> ph  <->  [.]  -.  ph )
 
Theoremnotal 24991 It's false that  ph always holds iff  -.  ph eventually holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( -.  [.] ph  <->  <>  -.  ph )
 
Theoremltl4ev 24992 The contrapositive of ax-ltl4 24977. If the truth of  ph in each step implies it is true in the previous step, and  ph is eventually true, then  ph is true in the first step. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  (
 ( [.] ( () ph  -> 
 ph )  /\  <> ph )  -> 
 ph )
 
Axiomax-ltl5 24993  ph holds until  ps iff  ps holds in the current step or  ph holds in the current step and in the next step  ph holds until  ps. (Contributed by FL, 27-Feb-2011.)
 |-  (
 ( ph  until  ps )  <->  ( ps  \/  ( ph  /\ 
 () ( ph  until  ps )
 ) ) )
 
Axiomax-ltl6 24994 If  ph holds until  ps then eventually  ps holds. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( ph  until  ps )  -> 
 <> ps )
 
Theoremnopsthph 24995 If  ps doesn't hold in the first step and  ph holds until  ps then  ph holds. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  (
 ( -.  ps  /\  ( ph  until  ps ) )  ->  ph )
 
Theoremphthps 24996 If  ph doesn't hold in the current step and  ph holds until  ps then  ps holds in the current step. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( -.  ph  /\  ( ph  until  ps ) )  ->  ps )
 
Theoremimunt 24997 If  ps is true, then  ph is true until  ps. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( ps  ->  ( ph  until  ps )
 )
 
Theoremevpexun 24998 Eventually  ph expressed with the  until operator. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Mario Carneiro, 30-Aug-2016.)
 |-  ( <> ph  <->  (  T.  until  ph ) )
 
Theoremalbineal 24999  ph always holds iff  ph holds in the first step and always holds in the next step. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] ph  <->  ( ph  /\  () [.] ph ) )
 
Theoremalneal1 25000 If  ph always holds, it holds in the first step. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] ph  ->  ph )
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