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Theorem List for Metamath Proof Explorer - 24101-24200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremintn3an1d 24101 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 24102 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 24103 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 24104 A consequence of  /\ associativity in a triple conjunct. (Contributed by FL, 14-Jul-2007.)
 |-  (
 ( ph  /\  ps  /\  ( ch  /\  th )
 ) 
 <->  ( ( ph  /\  ps  /\ 
 ch )  /\  th ) )
 
Theoremand4com 24105 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 24106 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 24107 Proof by contradiction combined with a disjunction. (Contributed by FL, 20-Apr-2011.)
 |-  ( ph  ->  ps )   &    |-  ( -.  ph  ->  ch )   =>    |-  ( ps  \/  ch )
 
Theoremcondisd 24108 Proof by contradiction combined with a disjunction. (Contributed by FL, 20-Apr-2011.)
 |-  (
 ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 -.  ps )  ->  th )   =>    |-  ( ph  ->  ( ch  \/  th ) )
 
Theoremeeeeanv 24109* 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 24110 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 24111 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 24112 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 24113 Alternate definition of true. In fact any tautology is a definition of true. (Contributed by FL, 23-Mar-2011.)
 |-  (  T. 
 <->  ( ph  \/  -.  ph ) )
 
Theoremtrant 24114 A true antecedent can be removed. (Contributed by FL, 16-Apr-2012.)
 |-  (
 (  T.  ->  ph )  <->  ph )
 
Theoremvutr 24115 Vacuous universal quantification is true. (Contributed by FL, 16-Apr-2012.)
 |-  (  T. 
 <-> 
 A. x  e.  (/)  ph )
 
Theoremtrcrm 24116 True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.)
 |-  (
 (  T.  /\  ph )  <->  ph )
 
Theoremtnf 24117 True is not false. (Contributed by FL, 20-Mar-2011.)
 |-  (  T. 
 <->  -.  F.  )
 
Theoremfacrm 24118 False can be removed from a disjunction. (Contributed by FL, 20-Mar-2011.)
 |-  (
 (  F.  \/  ph )  <->  ph )
 
Theoremfordisxex 24119 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 24120* 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 24121* 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 24122* 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 24123* 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 24124* Removal of an universal restricted quantifier in an antecedent. See also reximdva0 3373. (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 24125* 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 24126* 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 24127* 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 )
 ) )
 
Theoremrcla42edv 24128* 2-variable restricted existential specialization, using implicit substitution. (rcla42ev 2829 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 24129 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 24130* A variable introduction law for ordered triples. See eqvinop 4144. (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 24131* Substitution of class  A for ordered triple  <. <. x ,  y >. ,  z
>.. See copsexg 4145. (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 24132 Deduction adding difference to the right in a class equality. (Moved into main set.mm as difeq12d 3212 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 24133 Inequality. (Contributed by FL, 30-May-2014.)
 |-  A  =/=  B   &    |-  A  =  C   &    |-  B  =  D   =>    |-  C  =/=  D
 
Theoremsbcbidv2 24134* 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 ) )
 
16.11.3  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 24139, ax-ltl2 24140, ax-ltl3 24141, ax-ltl4 , ax-lmp 24143, and ax-nmp 24144. 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 24135 An always true proposition is well formed.
 wff  [.] ph
 
Syntaxwdia 24136 An eventually true proposition is well formed.
 wff  <> ph
 
Syntaxwcirc 24137 A proposition true in the next step is well formed.
 wff  ()
 ph
 
Syntaxwunt 24138 The proposition " ph is true until  ps is true " is well formed.
 wff  ( ph  until  ps )
 
Axiomax-ltl1 24139 If  ( ph  ->  ps ) and  ph always hold then  ps always holds. (Contributed by FL, 22-Feb-2011.)
 |-  ( [.] ( ph  ->  ps )  ->  ( [.] ph  ->  [.]
 ps ) )
 
Axiomax-ltl2 24140  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 24141 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 24142 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 24143 If  ph is a theorem then it always holds. (Contributed by FL, 22-Feb-2011.)
 |-  ph   =>    |- 
 [.] ph
 
Axiomax-nmp 24144 If  ph is a theorem then it holds in the next step. (Contributed by FL, 20-Mar-2011.)
 |-  ph   =>    |- 
 () ph
 
Definitiondf-dia 24145  ph eventually holds iff it is not true that  -.  ph always holds. (Contributed by FL, 22-Feb-2011.)
 |-  ( <> ph  <->  -. 
 [.]  -.  ph )
 
Theoremimpbox 24146 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 24147 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 24148 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 24149 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 24150  ( 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 24151  ( 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 24152  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 24153 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 24154 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 24155 It's false that  ph eventually holds iff  -.  ph always holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( -. 
 <> ph  <->  [.]  -.  ph )
 
Theoremnotal 24156 It's false that  ph always holds iff  -.  ph eventually holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( -.  [.] ph  <->  <>  -.  ph )
 
Theoremltl4ev 24157 The contrapositive of ax-ltl4 24142. 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 24158  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 24159 If  ph holds until  ps then eventually  ps holds. (Contributed by FL, 20-Mar-2011.)
 |-  (
 ( ph  until  ps )  -> 
 <> ps )
 
Theoremnopsthph 24160 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 24161 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 24162 If  ps is true, then  ph is true until  ps. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( ps  ->  ( ph  until  ps )
 )
 
Theoremevpexun 24163 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 24164  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 24165 If  ph always holds, it holds in the first step. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] ph  ->  ph )
 
Theoremalneal2 24166 If  ph always holds, it always holds in the next step. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] ph  ->  () [.] ph )
 
Theoremalne 24167 If  ph always holds, it holds in the next step. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] ph  ->  () ph )
 
Theoremalalifal 24168 It is always true that  ph always holds iff 
ph always holds. (Contributed by FL, 20-Mar-2011.)
 |-  ( [.] [.] ph  <->  [.] ph )
 
Theoremalneal1a 24169 Removing a box in the consequent. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  ( [.] ph  ->  ps )   =>    |-  ( [.] ph  ->  [.] ps )
 
Theoremimpbox2 24170 Removing boxes in the antecedents and consequent. (Contributed by FL, 16-Sep-2016.)
 |-  ( ch  ->  ( ph  ->  ps ) )   =>    |-  ( [.] ch  ->  ( [.] ph  ->  [.] ps ) )
 
Theoremboxand 24171 Distributivity of  [.] over  /\. (Contributed by FL, 1-Sep-2016.)
 |-  ( [.] ( ph  /\  ps ) 
 <->  ( [.] ph  /\  [.] ps ) )
 
Theoremboxrim 24172 If  [.] ph implies  ps in the current world, then it implies  ps in every world. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ps )   =>    |-  ( [.] ph  ->  [.] ps )
 
Theoremboximd 24173 Distribute 'always' over implication when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( [.] ps  ->  [.] ch ) )
 
Theoremnxtimd 24174 Distribute 'next' over implication when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( () ps  ->  () ch ) )
 
Theoremdiaimd 24175 Distribute 'eventually' over implication when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  (
 <> ps  ->  <> ch ) )
 
Theoremboxbid 24176 Distribute 'always' over biconditional when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  <->  ch ) )   =>    |-  ( [.] ph  ->  ( [.] ps  <->  [.] ch ) )
 
Theoremnxtbid 24177 Distribute 'next' over biconditional when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  <->  ch ) )   =>    |-  ( [.] ph  ->  ( () ps  <->  () ch ) )
 
Theoremdiabid 24178 Distribute 'eventually' over biconditional when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  <->  ch ) )   =>    |-  ( [.] ph  ->  (
 <> ps  <->  <> ch ) )
 
Theoremevevifev 24179 It is eventually true that  ph eventually holds iff  ph eventually holds. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( <> <> ph  <->  <> ph )
 
Theoremalthalne 24180 If  ph is always true then it is always true that  ph holds in the next step. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  [.] () ph )
 
Theoremtrtrst 24181  T. is true in every step. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Mario Carneiro, 30-Aug-2016.)
 |-  [.]  T.
 
Theoremunttr 24182 It's true that  ph is true until true is true. (Contributed by FL, 27-Feb-2011.) (Proof shortened by Mario Carneiro, 30-Aug-2016.)
 |-  ( ph  until  T.  )
 
Theoremuntind 24183 An "induction principle" for until, roughly stating that it is the least fixed point satisfying a property like ax-ltl5 24158. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  ->  ( ( ph  until  ps )  ->  th ) )
 
Theoremuntindd 24184 An "induction principle" for until, roughly stating that it is the least fixed point satisfying a property like ax-ltl5 24158. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( ps  ->  th )   &    |-  ( ( ph  /\ 
 () th )  ->  th )   =>    |-  (
 ( ph  until  ps )  ->  th )
 
Theoremuntim1d 24185 Congruence axiom for until. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( ( ps  until  th )  ->  ( ch  until  th )
 ) )
 
Theoremuntim2d 24186 Congruence axiom for until. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  ->  ch ) )   =>    |-  ( [.] ph  ->  ( ( th  until  ps )  ->  ( th  until  ch )
 ) )
 
Theoremuntbi12d 24187 Congruence axiom for until. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( [.] ph  ->  ( ps  <->  ch ) )   &    |-  ( [.] ph  ->  ( th  <->  ta ) )   =>    |-  ( [.] ph  ->  ( ( ps  until  th )  <->  ( ch  until  ta ) ) )
 
Theoremuntbi12i 24188 Congruence axiom for until. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( ph 
 <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  until  ch )  <->  ( ps  until  th ) )
 
Theoremaxlmp1 24189 If  ph always holds then it is a theorem. (Contributed by FL, 16-Sep-2016.)
 |-  [.] ph   =>    |-  ph
 
Theoremaxlmp2 24190 Moving a universal quantifier inside and outside a box. (Contributed by FL, 16-Sep-2016.)
 |-  A. x [.] ph   =>    |- 
 [.] A. x ph
 
Theoremaxlmp3 24191 Moving a universal quantifier inside and outside a box. (Contributed by FL, 16-Sep-2016.)
 |-  [.] A. x ph   =>    |- 
 A. x [.] ph
 
Axiomax-lll 24192 Set equality is true in all worlds. (Contributed by Mario Carneiro, 30-Aug-2016.)
 |-  ( x  =  y  ->  [.] x  =  y )
 
Theoremaxlll2 24193 One can add or remove a box in front of  x  =  y. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  ( [.] x  =  y  <->  x  =  y
 )
 
Theoremcdeqbox 24194 Distribute conditional equality over 'always'. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph 
 <->  ps ) )   =>    |- CondEq ( x  =  y  ->  ( [.] ph  <->  [.]
 ps ) )
 
Theoremcdeqnxt 24195 Distribute conditional equality over 'next'. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph 
 <->  ps ) )   =>    |- CondEq ( x  =  y  ->  ( () ph  <->  ()
 ps ) )
 
Theoremcdequnt 24196 Distribute conditional equality over 'until'. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- CondEq ( x  =  y  ->  ( ph 
 <->  ps ) )   &    |- CondEq ( x  =  y  ->  ( ch 
 <-> 
 th ) )   =>    |- CondEq ( x  =  y  ->  ( ( ph  until  ch )  <->  ( ps  until  th ) ) )
 
16.11.4  Operations
 
Theoremssoprab2g 24197* Inference of operation class abstraction subclass from implication. (Contributed by FL, 24-Jan-2010.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  { <. <. x ,  y >. ,  z >.  |  ps } 
 C_  { <. <. x ,  y >. ,  z >.  |  ch } )
 
Theoremdmoprabsss 24198* The domain of an operation class abstraction. Compare dmoprabss 5781. (Contributed by FL, 24-Jan-2010.)
 |-  dom  {
 <. <. x ,  y >. ,  z >.  |  (
 <. x ,  y >.  e.  ( A  X.  B )  /\  ph ) }  C_  ( A  X.  B )
 
Theoremoprssopvg 24199 Value returned by the operation  G in terms of the value returned by the "super"-operation  F. (A version of oprssov 5841 adapted to partial operations.) (Contributed by FL, 5-Oct-2009.)
 |-  (
 ( Fun  F  /\  G  C_  F  /\  <. A ,  B >.  e.  dom  G )  ->  ( A F B )  =  ( A G B ) )
 
Theoremdmoprabss6 24200* The domain of an operation class abstraction. (A version of dmoprabss 5781 adapted to partial operations.) (Contributed by FL, 18-Apr-2010.)
 |-  B  e.  C   =>    |-  ( Rel  A  ->  dom  { <. <. x ,  y >. ,  z >.  |  (
 <. x ,  y >.  e.  A  /\  z  =  B ) }  =  A )
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