Nuprl Lemma : l_all_cons
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀x:T. ∀L:T List. ((∀y∈[x / L].P[y]) ⇐⇒ P[x] ∧ (∀y∈L.P[y]))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
cons: [a / b],
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
rev_implies: P ⇐ Q,
or: P ∨ Q,
guard: {T},
subtype_rel: A ⊆r B
Lemmas referenced :
l_all_iff,
cons_wf,
l_member_wf,
cons_member,
equal_wf,
l_all_wf,
list_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
dependent_functionElimination,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
setElimination,
rename,
setEquality,
productElimination,
independent_functionElimination,
inlFormation,
inrFormation,
productEquality,
universeEquality,
functionEquality,
unionElimination,
equalitySymmetry,
dependent_set_memberEquality,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}x:T. \mforall{}L:T List. ((\mforall{}y\mmember{}[x / L].P[y]) \mLeftarrow{}{}\mRightarrow{} P[x] \mwedge{} (\mforall{}y\mmember{}L.P[y]))
Date html generated:
2016_10_21-AM-09_49_04
Last ObjectModification:
2016_07_12-AM-05_08_43
Theory : list_0
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