Nuprl Lemma : l_all_functionality
∀[T:Type]. ∀L:T List. ∀P,Q:T ⟶ ℙ.  ((∀x:T. ((x ∈ L) 
⇒ (P[x] 
⇐⇒ Q[x]))) 
⇒ {(∀x∈L.P[x]) 
⇐⇒ (∀x∈L.Q[x])})
Proof
Definitions occuring in Statement : 
l_all: (∀x∈L.P[x])
, 
l_member: (x ∈ l)
, 
list: T List
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
guard: {T}
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
implies: P 
⇒ Q
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
l_all: (∀x∈L.P[x])
, 
member: t ∈ T
, 
int_seg: {i..j-}
, 
uimplies: b supposing a
, 
sq_stable: SqStable(P)
, 
lelt: i ≤ j < k
, 
squash: ↓T
, 
prop: ℙ
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
rev_implies: P 
⇐ Q
Lemmas referenced : 
list_wf, 
iff_wf, 
all_wf, 
l_member_wf, 
l_all_wf, 
length_wf, 
int_seg_wf, 
select_member, 
sq_stable__le, 
select_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
lambdaFormation, 
independent_pairFormation, 
sqequalHypSubstitution, 
cut, 
hypothesis, 
dependent_functionElimination, 
thin, 
hypothesisEquality, 
lemma_by_obid, 
isectElimination, 
cumulativity, 
setElimination, 
rename, 
independent_isectElimination, 
natural_numberEquality, 
independent_functionElimination, 
introduction, 
productElimination, 
sqequalRule, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
lambdaEquality, 
applyEquality, 
setEquality, 
functionEquality, 
universeEquality
Latex:
\mforall{}[T:Type]
    \mforall{}L:T  List.  \mforall{}P,Q:T  {}\mrightarrow{}  \mBbbP{}.    ((\mforall{}x:T.  ((x  \mmember{}  L)  {}\mRightarrow{}  (P[x]  \mLeftarrow{}{}\mRightarrow{}  Q[x])))  {}\mRightarrow{}  \{(\mforall{}x\mmember{}L.P[x])  \mLeftarrow{}{}\mRightarrow{}  (\mforall{}x\mmember{}L.Q[x])\})
Date html generated:
2016_05_14-AM-06_40_39
Last ObjectModification:
2016_01_14-PM-08_19_53
Theory : list_0
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