Nuprl Lemma : decdr-to-bool_wf
∀[T:Type]. ∀[A,B:T ⟶ ℙ]. ∀[d:x:T ⟶ (A[x] + B[x])].  (bool(d) ∈ T ⟶ 𝔹)
Proof
Definitions occuring in Statement : 
decdr-to-bool: bool(d)
, 
bool: 𝔹
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
so_apply: x[s]
, 
member: t ∈ T
, 
function: x:A ⟶ B[x]
, 
union: left + right
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
so_apply: x[s]
, 
subtype_rel: A ⊆r B
, 
prop: ℙ
, 
decdr-to-bool: bool(d)
, 
it: ⋅
, 
bfalse: ff
, 
btrue: tt
, 
ifthenelse: if b then t else f fi 
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
Lemmas referenced : 
subtype_rel_self, 
btrue_wf, 
bfalse_wf, 
equal_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
cut, 
sqequalHypSubstitution, 
hypothesis, 
functionEquality, 
hypothesisEquality, 
unionEquality, 
applyEquality, 
thin, 
sqequalRule, 
instantiate, 
introduction, 
extract_by_obid, 
isectElimination, 
because_Cache, 
cumulativity, 
universeEquality, 
lambdaEquality, 
lambdaFormation, 
equalityTransitivity, 
equalitySymmetry, 
unionElimination, 
dependent_functionElimination, 
independent_functionElimination
Latex:
\mforall{}[T:Type].  \mforall{}[A,B:T  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[d:x:T  {}\mrightarrow{}  (A[x]  +  B[x])].    (bool(d)  \mmember{}  T  {}\mrightarrow{}  \mBbbB{})
Date html generated:
2019_10_30-AM-07_36_05
Last ObjectModification:
2018_08_21-PM-02_01_15
Theory : reals
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