Nuprl Lemma : apply-ifthenelse
∀[T,U:Type]. ∀[f:T ⟶ U]. ∀[b:𝔹]. ∀[x,y:T].  (f[if b then x else y fi ] ~ if b then f[x] else f[y] fi )
Proof
Definitions occuring in Statement : 
ifthenelse: if b then t else f fi 
, 
bool: 𝔹
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s]
, 
function: x:A ⟶ B[x]
, 
universe: Type
, 
sqequal: s ~ t
Definitions unfolded in proof : 
bool: 𝔹
, 
ifthenelse: if b then t else f fi 
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
Lemmas referenced : 
bool_wf
Rules used in proof : 
sqequalHypSubstitution, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
unionElimination, 
thin, 
sqequalRule, 
hypothesisEquality, 
because_Cache, 
cut, 
lemma_by_obid, 
hypothesis, 
functionEquality, 
universeEquality, 
isect_memberFormation, 
introduction, 
sqequalAxiom, 
isect_memberEquality, 
isectElimination
Latex:
\mforall{}[T,U:Type].  \mforall{}[f:T  {}\mrightarrow{}  U].  \mforall{}[b:\mBbbB{}].  \mforall{}[x,y:T].
    (f[if  b  then  x  else  y  fi  ]  \msim{}  if  b  then  f[x]  else  f[y]  fi  )
Date html generated:
2016_05_13-PM-04_01_23
Last ObjectModification:
2015_12_26-AM-10_49_03
Theory : bool_1
Home
Index