Nuprl Lemma : l_all_fwd
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀L:T List. ∀x:T. ((x ∈ L) ⇒ (∀y∈L.P[y]) ⇒ P[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,
so_apply: x[s],
all: ∀x:A. B[x],
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,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
guard: {T}
Lemmas referenced :
l_all_iff,
l_member_wf,
l_all_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
dependent_functionElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
hypothesis,
setEquality,
productElimination,
independent_functionElimination,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}L:T List. \mforall{}x:T. ((x \mmember{} L) {}\mRightarrow{} (\mforall{}y\mmember{}L.P[y]) {}\mRightarrow{} P[x])
Date html generated:
2019_06_20-PM-01_24_43
Last ObjectModification:
2018_08_24-PM-10_48_27
Theory : list_1
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