Nuprl Lemma : assert-bl-exists2
∀[T:Type]. ∀L:T List. ∀P:{x:T| (x ∈ L)} ⟶ 𝔹. (↑(∃x∈L.P[x])_b ⇐⇒ ∃x:T. ((x ∈ L) ∧ (↑P[x])))
Proof
Definitions occuring in Statement :
bl-exists: (∃x∈L.P[x])_b,
l_member: (x ∈ l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
rev_implies: P ⇐ Q
Lemmas referenced :
assert-bl-exists,
l_exists_iff,
assert_wf,
l_member_wf,
bl-exists_wf,
exists_wf,
bool_wf,
list_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
productElimination,
independent_pairFormation,
independent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
setEquality,
cumulativity,
productEquality,
because_Cache,
dependent_set_memberEquality,
promote_hyp,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}. (\muparrow{}(\mexists{}x\mmember{}L.P[x])\_b \mLeftarrow{}{}\mRightarrow{} \mexists{}x:T. ((x \mmember{} L) \mwedge{} (\muparrow{}P[x])))
Date html generated:
2016_05_14-PM-02_10_13
Last ObjectModification:
2015_12_26-PM-05_04_40
Theory : list_1
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