Nuprl Lemma : l-all-iff
∀[T:Type]. ∀L:T List. ∀[P:{x:T| (x ∈ L)} ⟶ ℙ]. (∀x∈L.P[x] ⇐⇒ (∀x∈L.P[x]))
Proof
Definitions occuring in Statement :
l-all: ∀x∈L.P[x],
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
and: P ∧ Q,
l-all: ∀x∈L.P[x],
top: Top,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
not: ¬A,
false: False
Lemmas referenced :
list_induction,
uall_wf,
iff_wf,
l-all_wf,
l_member_wf,
l_all_wf2,
list_wf,
reduce_nil_lemma,
l_all_nil,
true_wf,
nil_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
reduce_cons_lemma,
subtype_rel_self,
l_all_cons,
cons_wf,
list-subtype
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
universeEquality,
applyEquality,
setElimination,
rename,
hypothesis,
setEquality,
because_Cache,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
natural_numberEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
productEquality,
addLevel
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbP{}]. (\mforall{}x\mmember{}L.P[x] \mLeftarrow{}{}\mRightarrow{} (\mforall{}x\mmember{}L.P[x]))
Date html generated:
2019_10_15-AM-11_08_59
Last ObjectModification:
2018_08_25-PM-00_08_55
Theory : general
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